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filter-thms-collapse.agda
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filter-thms-collapse.agda
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module filter-thms where
open import prelude
open import functions
open import quaternary-semantics
open import quaternary-thms
open import filter-semantics
⊙F-zerol : ∀{x} → (zero ⊙F x) ≤₄ zero
⊙F-zerol {zero} = triv
⊙F-zerol {forth} = triv
⊙F-zerol {half} = triv
⊙F-zerol {one} = triv
⊙F-zeror : ∀{x} → (x ⊙F zero) ≤₄ zero
⊙F-zeror {zero} = triv
⊙F-zeror {forth} = triv
⊙F-zeror {half} = triv
⊙F-zeror {one} = triv
⊙F-contract : (∀{a} → (a ⊙F a) ≡ a) → ⊥ {lzero}
⊙F-contract p with p {half}
... | ()
⊙F-sym : ∀{a b} → a ⊙F b ≡ b ⊙F a
⊙F-sym {zero} {zero} = refl
⊙F-sym {zero} {forth} = refl
⊙F-sym {zero} {half} = refl
⊙F-sym {zero} {one} = refl
⊙F-sym {forth} {zero} = refl
⊙F-sym {forth} {forth} = refl
⊙F-sym {forth} {half} = refl
⊙F-sym {forth} {one} = refl
⊙F-sym {half} {zero} = refl
⊙F-sym {half} {forth} = refl
⊙F-sym {half} {half} = refl
⊙F-sym {half} {one} = refl
⊙F-sym {one} {zero} = refl
⊙F-sym {one} {forth} = refl
⊙F-sym {one} {half} = refl
⊙F-sym {one} {one} = refl
⊙F-assoc : ∀{a b c} → (a ⊙F b) ⊙F c ≡ a ⊙F (b ⊙F c)
⊙F-assoc {zero} {zero} {zero} = refl
⊙F-assoc {zero} {zero} {forth} = refl
⊙F-assoc {zero} {zero} {half} = refl
⊙F-assoc {zero} {zero} {one} = refl
⊙F-assoc {zero} {forth} {zero} = refl
⊙F-assoc {zero} {forth} {forth} = refl
⊙F-assoc {zero} {forth} {half} = refl
⊙F-assoc {zero} {forth} {one} = refl
⊙F-assoc {zero} {half} {zero} = refl
⊙F-assoc {zero} {half} {forth} = refl
⊙F-assoc {zero} {half} {half} = refl
⊙F-assoc {zero} {half} {one} = refl
⊙F-assoc {zero} {one} {zero} = refl
⊙F-assoc {zero} {one} {forth} = refl
⊙F-assoc {zero} {one} {half} = refl
⊙F-assoc {zero} {one} {one} = refl
⊙F-assoc {forth} {zero} {zero} = refl
⊙F-assoc {forth} {zero} {forth} = refl
⊙F-assoc {forth} {zero} {half} = refl
⊙F-assoc {forth} {zero} {one} = refl
⊙F-assoc {forth} {forth} {zero} = refl
⊙F-assoc {forth} {forth} {forth} = refl
⊙F-assoc {forth} {forth} {half} = refl
⊙F-assoc {forth} {forth} {one} = refl
⊙F-assoc {forth} {half} {zero} = refl
⊙F-assoc {forth} {half} {forth} = refl
⊙F-assoc {forth} {half} {half} = refl
⊙F-assoc {forth} {half} {one} = refl
⊙F-assoc {forth} {one} {zero} = refl
⊙F-assoc {forth} {one} {forth} = refl
⊙F-assoc {forth} {one} {half} = refl
⊙F-assoc {forth} {one} {one} = refl
⊙F-assoc {half} {zero} {zero} = refl
⊙F-assoc {half} {zero} {forth} = refl
⊙F-assoc {half} {zero} {half} = refl
⊙F-assoc {half} {zero} {one} = refl
⊙F-assoc {half} {forth} {zero} = refl
⊙F-assoc {half} {forth} {forth} = refl
⊙F-assoc {half} {forth} {half} = refl
⊙F-assoc {half} {forth} {one} = refl
⊙F-assoc {half} {half} {zero} = refl
⊙F-assoc {half} {half} {forth} = refl
⊙F-assoc {half} {half} {half} = refl
⊙F-assoc {half} {half} {one} = refl
⊙F-assoc {half} {one} {zero} = refl
⊙F-assoc {half} {one} {forth} = refl
⊙F-assoc {half} {one} {half} = refl
⊙F-assoc {half} {one} {one} = refl
⊙F-assoc {one} {zero} {zero} = refl
⊙F-assoc {one} {zero} {forth} = refl
⊙F-assoc {one} {zero} {half} = refl
⊙F-assoc {one} {zero} {one} = refl
⊙F-assoc {one} {forth} {zero} = refl
⊙F-assoc {one} {forth} {forth} = refl
⊙F-assoc {one} {forth} {half} = refl
⊙F-assoc {one} {forth} {one} = refl
⊙F-assoc {one} {half} {zero} = refl
⊙F-assoc {one} {half} {forth} = refl
⊙F-assoc {one} {half} {half} = refl
⊙F-assoc {one} {half} {one} = refl
⊙F-assoc {one} {one} {zero} = refl
⊙F-assoc {one} {one} {forth} = refl
⊙F-assoc {one} {one} {half} = refl
⊙F-assoc {one} {one} {one} = refl
⊙F-func : {a c b d : Four} → a ≤₄ c → b ≤₄ d → (a ⊙F b) ≤₄ (c ⊙F d)
⊙F-func {zero} {zero} {zero} {zero} p₁ p₂ = triv
⊙F-func {zero} {zero} {zero} {forth} p₁ p₂ = triv
⊙F-func {zero} {zero} {zero} {half} p₁ p₂ = triv
⊙F-func {zero} {zero} {zero} {one} p₁ p₂ = triv
⊙F-func {zero} {zero} {forth} {zero} p₁ p₂ = triv
⊙F-func {zero} {zero} {forth} {forth} p₁ p₂ = triv
⊙F-func {zero} {zero} {forth} {half} p₁ p₂ = triv
⊙F-func {zero} {zero} {forth} {one} p₁ p₂ = triv
⊙F-func {zero} {zero} {half} {zero} p₁ p₂ = triv
⊙F-func {zero} {zero} {half} {forth} p₁ p₂ = triv
⊙F-func {zero} {zero} {half} {half} p₁ p₂ = triv
⊙F-func {zero} {zero} {half} {one} p₁ p₂ = triv
⊙F-func {zero} {zero} {one} {zero} p₁ p₂ = triv
⊙F-func {zero} {zero} {one} {forth} p₁ p₂ = triv
⊙F-func {zero} {zero} {one} {half} p₁ p₂ = triv
⊙F-func {zero} {zero} {one} {one} p₁ p₂ = triv
⊙F-func {zero} {forth} {zero} {zero} p₁ p₂ = triv
⊙F-func {zero} {forth} {zero} {forth} p₁ p₂ = triv
⊙F-func {zero} {forth} {zero} {half} p₁ p₂ = triv
⊙F-func {zero} {forth} {zero} {one} p₁ p₂ = triv
⊙F-func {zero} {forth} {forth} {zero} p₁ p₂ = triv
⊙F-func {zero} {forth} {forth} {forth} p₁ p₂ = triv
⊙F-func {zero} {forth} {forth} {half} p₁ p₂ = triv
⊙F-func {zero} {forth} {forth} {one} p₁ p₂ = triv
⊙F-func {zero} {forth} {half} {zero} p₁ p₂ = triv
⊙F-func {zero} {forth} {half} {forth} p₁ p₂ = triv
⊙F-func {zero} {forth} {half} {half} p₁ p₂ = triv
⊙F-func {zero} {forth} {half} {one} p₁ p₂ = triv
⊙F-func {zero} {forth} {one} {zero} p₁ p₂ = triv
⊙F-func {zero} {forth} {one} {forth} p₁ p₂ = triv
⊙F-func {zero} {forth} {one} {half} p₁ p₂ = triv
⊙F-func {zero} {forth} {one} {one} p₁ p₂ = triv
⊙F-func {zero} {half} {zero} {zero} p₁ p₂ = triv
⊙F-func {zero} {half} {zero} {forth} p₁ p₂ = triv
⊙F-func {zero} {half} {zero} {half} p₁ p₂ = triv
⊙F-func {zero} {half} {zero} {one} p₁ p₂ = triv
⊙F-func {zero} {half} {forth} {zero} p₁ p₂ = triv
⊙F-func {zero} {half} {forth} {forth} p₁ p₂ = triv
⊙F-func {zero} {half} {forth} {half} p₁ p₂ = triv
⊙F-func {zero} {half} {forth} {one} p₁ p₂ = triv
⊙F-func {zero} {half} {half} {zero} p₁ p₂ = triv
⊙F-func {zero} {half} {half} {forth} p₁ p₂ = triv
⊙F-func {zero} {half} {half} {half} p₁ p₂ = triv
⊙F-func {zero} {half} {half} {one} p₁ p₂ = triv
⊙F-func {zero} {half} {one} {zero} p₁ p₂ = triv
⊙F-func {zero} {half} {one} {forth} p₁ p₂ = triv
⊙F-func {zero} {half} {one} {half} p₁ p₂ = triv
⊙F-func {zero} {half} {one} {one} p₁ p₂ = triv
⊙F-func {zero} {one} {zero} {zero} p₁ p₂ = triv
⊙F-func {zero} {one} {zero} {forth} p₁ p₂ = triv
⊙F-func {zero} {one} {zero} {half} p₁ p₂ = triv
⊙F-func {zero} {one} {zero} {one} p₁ p₂ = triv
⊙F-func {zero} {one} {forth} {zero} p₁ p₂ = triv
⊙F-func {zero} {one} {forth} {forth} p₁ p₂ = triv
⊙F-func {zero} {one} {forth} {half} p₁ p₂ = triv
⊙F-func {zero} {one} {forth} {one} p₁ p₂ = triv
⊙F-func {zero} {one} {half} {zero} p₁ p₂ = triv
⊙F-func {zero} {one} {half} {forth} p₁ p₂ = triv
⊙F-func {zero} {one} {half} {half} p₁ p₂ = triv
⊙F-func {zero} {one} {half} {one} p₁ p₂ = triv
⊙F-func {zero} {one} {one} {zero} p₁ p₂ = triv
⊙F-func {zero} {one} {one} {forth} p₁ p₂ = triv
⊙F-func {zero} {one} {one} {half} p₁ p₂ = triv
⊙F-func {zero} {one} {one} {one} p₁ p₂ = triv
⊙F-func {forth} {zero} {zero} {zero} p₁ p₂ = triv
⊙F-func {forth} {zero} {zero} {forth} p₁ p₂ = triv
⊙F-func {forth} {zero} {zero} {half} p₁ p₂ = triv
⊙F-func {forth} {zero} {zero} {one} p₁ p₂ = triv
⊙F-func {forth} {zero} {forth} {zero} () ()
⊙F-func {forth} {zero} {forth} {forth} () p₂
⊙F-func {forth} {zero} {forth} {half} () p₂
⊙F-func {forth} {zero} {forth} {one} () p₂
⊙F-func {forth} {zero} {half} {zero} () ()
⊙F-func {forth} {zero} {half} {forth} () ()
⊙F-func {forth} {zero} {half} {half} () p₂
⊙F-func {forth} {zero} {half} {one} () p₂
⊙F-func {forth} {zero} {one} {zero} () ()
⊙F-func {forth} {zero} {one} {forth} () ()
⊙F-func {forth} {zero} {one} {half} () ()
⊙F-func {forth} {zero} {one} {one} () p₂
⊙F-func {forth} {forth} {zero} {zero} p₁ p₂ = triv
⊙F-func {forth} {forth} {zero} {forth} p₁ p₂ = triv
⊙F-func {forth} {forth} {zero} {half} p₁ p₂ = triv
⊙F-func {forth} {forth} {zero} {one} p₁ p₂ = triv
⊙F-func {forth} {forth} {forth} {zero} p₁ ()
⊙F-func {forth} {forth} {forth} {forth} p₁ p₂ = triv
⊙F-func {forth} {forth} {forth} {half} p₁ p₂ = triv
⊙F-func {forth} {forth} {forth} {one} p₁ p₂ = triv
⊙F-func {forth} {forth} {half} {zero} p₁ ()
⊙F-func {forth} {forth} {half} {forth} p₁ p₂ = triv
⊙F-func {forth} {forth} {half} {half} p₁ p₂ = triv
⊙F-func {forth} {forth} {half} {one} p₁ p₂ = triv
⊙F-func {forth} {forth} {one} {zero} p₁ ()
⊙F-func {forth} {forth} {one} {forth} p₁ p₂ = triv
⊙F-func {forth} {forth} {one} {half} p₁ p₂ = triv
⊙F-func {forth} {forth} {one} {one} p₁ p₂ = triv
⊙F-func {forth} {half} {zero} {zero} p₁ p₂ = triv
⊙F-func {forth} {half} {zero} {forth} p₁ p₂ = triv
⊙F-func {forth} {half} {zero} {half} p₁ p₂ = triv
⊙F-func {forth} {half} {zero} {one} p₁ p₂ = triv
⊙F-func {forth} {half} {forth} {zero} p₁ ()
⊙F-func {forth} {half} {forth} {forth} p₁ p₂ = triv
⊙F-func {forth} {half} {forth} {half} p₁ p₂ = triv
⊙F-func {forth} {half} {forth} {one} p₁ p₂ = triv
⊙F-func {forth} {half} {half} {zero} p₁ ()
⊙F-func {forth} {half} {half} {forth} p₁ p₂ = triv
⊙F-func {forth} {half} {half} {half} p₁ p₂ = triv
⊙F-func {forth} {half} {half} {one} p₁ p₂ = triv
⊙F-func {forth} {half} {one} {zero} p₁ ()
⊙F-func {forth} {half} {one} {forth} p₁ p₂ = triv
⊙F-func {forth} {half} {one} {half} p₁ p₂ = triv
⊙F-func {forth} {half} {one} {one} p₁ p₂ = triv
⊙F-func {forth} {one} {zero} {zero} p₁ p₂ = triv
⊙F-func {forth} {one} {zero} {forth} p₁ p₂ = triv
⊙F-func {forth} {one} {zero} {half} p₁ p₂ = triv
⊙F-func {forth} {one} {zero} {one} p₁ p₂ = triv
⊙F-func {forth} {one} {forth} {zero} p₁ ()
⊙F-func {forth} {one} {forth} {forth} p₁ p₂ = triv
⊙F-func {forth} {one} {forth} {half} p₁ p₂ = triv
⊙F-func {forth} {one} {forth} {one} p₁ p₂ = triv
⊙F-func {forth} {one} {half} {zero} p₁ ()
⊙F-func {forth} {one} {half} {forth} p₁ p₂ = triv
⊙F-func {forth} {one} {half} {half} p₁ p₂ = triv
⊙F-func {forth} {one} {half} {one} p₁ p₂ = triv
⊙F-func {forth} {one} {one} {zero} p₁ ()
⊙F-func {forth} {one} {one} {forth} p₁ p₂ = triv
⊙F-func {forth} {one} {one} {half} p₁ p₂ = triv
⊙F-func {forth} {one} {one} {one} p₁ p₂ = triv
⊙F-func {half} {zero} {zero} {zero} p₁ p₂ = triv
⊙F-func {half} {zero} {zero} {forth} p₁ p₂ = triv
⊙F-func {half} {zero} {zero} {half} p₁ p₂ = triv
⊙F-func {half} {zero} {zero} {one} p₁ p₂ = triv
⊙F-func {half} {zero} {forth} {zero} () ()
⊙F-func {half} {zero} {forth} {forth} () p₂
⊙F-func {half} {zero} {forth} {half} () p₂
⊙F-func {half} {zero} {forth} {one} () p₂
⊙F-func {half} {zero} {half} {zero} () ()
⊙F-func {half} {zero} {half} {forth} () ()
⊙F-func {half} {zero} {half} {half} () p₂
⊙F-func {half} {zero} {half} {one} () p₂
⊙F-func {half} {zero} {one} {zero} () ()
⊙F-func {half} {zero} {one} {forth} () ()
⊙F-func {half} {zero} {one} {half} () ()
⊙F-func {half} {zero} {one} {one} () p₂
⊙F-func {half} {forth} {zero} {zero} p₁ p₂ = triv
⊙F-func {half} {forth} {zero} {forth} p₁ p₂ = triv
⊙F-func {half} {forth} {zero} {half} p₁ p₂ = triv
⊙F-func {half} {forth} {zero} {one} p₁ p₂ = triv
⊙F-func {half} {forth} {forth} {zero} () ()
⊙F-func {half} {forth} {forth} {forth} p₁ p₂ = triv
⊙F-func {half} {forth} {forth} {half} p₁ p₂ = triv
⊙F-func {half} {forth} {forth} {one} p₁ p₂ = triv
⊙F-func {half} {forth} {half} {zero} () ()
⊙F-func {half} {forth} {half} {forth} p₁ p₂ = triv
⊙F-func {half} {forth} {half} {half} p₁ p₂ = triv
⊙F-func {half} {forth} {half} {one} p₁ p₂ = triv
⊙F-func {half} {forth} {one} {zero} () ()
⊙F-func {half} {forth} {one} {forth} () ()
⊙F-func {half} {forth} {one} {half} () ()
⊙F-func {half} {forth} {one} {one} () p₂
⊙F-func {half} {half} {zero} {zero} p₁ p₂ = triv
⊙F-func {half} {half} {zero} {forth} p₁ p₂ = triv
⊙F-func {half} {half} {zero} {half} p₁ p₂ = triv
⊙F-func {half} {half} {zero} {one} p₁ p₂ = triv
⊙F-func {half} {half} {forth} {zero} p₁ ()
⊙F-func {half} {half} {forth} {forth} p₁ p₂ = triv
⊙F-func {half} {half} {forth} {half} p₁ p₂ = triv
⊙F-func {half} {half} {forth} {one} p₁ p₂ = triv
⊙F-func {half} {half} {half} {zero} p₁ ()
⊙F-func {half} {half} {half} {forth} p₁ p₂ = triv
⊙F-func {half} {half} {half} {half} p₁ p₂ = triv
⊙F-func {half} {half} {half} {one} p₁ p₂ = triv
⊙F-func {half} {half} {one} {zero} p₁ ()
⊙F-func {half} {half} {one} {forth} p₁ ()
⊙F-func {half} {half} {one} {half} p₁ ()
⊙F-func {half} {half} {one} {one} p₁ p₂ = triv
⊙F-func {half} {one} {zero} {zero} p₁ p₂ = triv
⊙F-func {half} {one} {zero} {forth} p₁ p₂ = triv
⊙F-func {half} {one} {zero} {half} p₁ p₂ = triv
⊙F-func {half} {one} {zero} {one} p₁ p₂ = triv
⊙F-func {half} {one} {forth} {zero} p₁ ()
⊙F-func {half} {one} {forth} {forth} p₁ p₂ = triv
⊙F-func {half} {one} {forth} {half} p₁ p₂ = triv
⊙F-func {half} {one} {forth} {one} p₁ p₂ = triv
⊙F-func {half} {one} {half} {zero} p₁ ()
⊙F-func {half} {one} {half} {forth} p₁ p₂ = triv
⊙F-func {half} {one} {half} {half} p₁ p₂ = triv
⊙F-func {half} {one} {half} {one} p₁ p₂ = triv
⊙F-func {half} {one} {one} {zero} p₁ ()
⊙F-func {half} {one} {one} {forth} p₁ ()
⊙F-func {half} {one} {one} {half} p₁ p₂ = triv
⊙F-func {half} {one} {one} {one} p₁ p₂ = triv
⊙F-func {one} {zero} {zero} {zero} p₁ p₂ = triv
⊙F-func {one} {zero} {zero} {forth} p₁ p₂ = triv
⊙F-func {one} {zero} {zero} {half} p₁ p₂ = triv
⊙F-func {one} {zero} {zero} {one} p₁ p₂ = triv
⊙F-func {one} {zero} {forth} {zero} () ()
⊙F-func {one} {zero} {forth} {forth} () p₂
⊙F-func {one} {zero} {forth} {half} () p₂
⊙F-func {one} {zero} {forth} {one} () p₂
⊙F-func {one} {zero} {half} {zero} () ()
⊙F-func {one} {zero} {half} {forth} () ()
⊙F-func {one} {zero} {half} {half} () p₂
⊙F-func {one} {zero} {half} {one} () p₂
⊙F-func {one} {zero} {one} {zero} () ()
⊙F-func {one} {zero} {one} {forth} () ()
⊙F-func {one} {zero} {one} {half} () ()
⊙F-func {one} {zero} {one} {one} () p₂
⊙F-func {one} {forth} {zero} {zero} p₁ p₂ = triv
⊙F-func {one} {forth} {zero} {forth} p₁ p₂ = triv
⊙F-func {one} {forth} {zero} {half} p₁ p₂ = triv
⊙F-func {one} {forth} {zero} {one} p₁ p₂ = triv
⊙F-func {one} {forth} {forth} {zero} () ()
⊙F-func {one} {forth} {forth} {forth} p₁ p₂ = triv
⊙F-func {one} {forth} {forth} {half} p₁ p₂ = triv
⊙F-func {one} {forth} {forth} {one} p₁ p₂ = triv
⊙F-func {one} {forth} {half} {zero} () ()
⊙F-func {one} {forth} {half} {forth} () ()
⊙F-func {one} {forth} {half} {half} () p₂
⊙F-func {one} {forth} {half} {one} () p₂
⊙F-func {one} {forth} {one} {zero} () ()
⊙F-func {one} {forth} {one} {forth} () ()
⊙F-func {one} {forth} {one} {half} () ()
⊙F-func {one} {forth} {one} {one} () p₂
⊙F-func {one} {half} {zero} {zero} p₁ p₂ = triv
⊙F-func {one} {half} {zero} {forth} p₁ p₂ = triv
⊙F-func {one} {half} {zero} {half} p₁ p₂ = triv
⊙F-func {one} {half} {zero} {one} p₁ p₂ = triv
⊙F-func {one} {half} {forth} {zero} () ()
⊙F-func {one} {half} {forth} {forth} p₁ p₂ = triv
⊙F-func {one} {half} {forth} {half} () p₂
⊙F-func {one} {half} {forth} {one} p₁ p₂ = triv
⊙F-func {one} {half} {half} {zero} () ()
⊙F-func {one} {half} {half} {forth} () ()
⊙F-func {one} {half} {half} {half} () p₂
⊙F-func {one} {half} {half} {one} p₁ p₂ = triv
⊙F-func {one} {half} {one} {zero} () ()
⊙F-func {one} {half} {one} {forth} () ()
⊙F-func {one} {half} {one} {half} () ()
⊙F-func {one} {half} {one} {one} () p₂
⊙F-func {one} {one} {zero} {zero} p₁ p₂ = triv
⊙F-func {one} {one} {zero} {forth} p₁ p₂ = triv
⊙F-func {one} {one} {zero} {half} p₁ p₂ = triv
⊙F-func {one} {one} {zero} {one} p₁ p₂ = triv
⊙F-func {one} {one} {forth} {zero} p₁ ()
⊙F-func {one} {one} {forth} {forth} p₁ p₂ = triv
⊙F-func {one} {one} {forth} {half} p₁ p₂ = triv
⊙F-func {one} {one} {forth} {one} p₁ p₂ = triv
⊙F-func {one} {one} {half} {zero} p₁ ()
⊙F-func {one} {one} {half} {forth} p₁ ()
⊙F-func {one} {one} {half} {half} p₁ p₂ = triv
⊙F-func {one} {one} {half} {one} p₁ p₂ = triv
⊙F-func {one} {one} {one} {zero} p₁ ()
⊙F-func {one} {one} {one} {forth} p₁ ()
⊙F-func {one} {one} {one} {half} p₁ ()
⊙F-func {one} {one} {one} {one} p₁ p₂ = triv
⊙F-distl : {a b c : Four} → (a ⊙F (b ⊔F c)) ≡ ((a ⊙F b) ⊔F (a ⊙F c))
⊙F-distl {zero} {zero} {zero} = refl
⊙F-distl {zero} {zero} {forth} = refl
⊙F-distl {zero} {zero} {half} = refl
⊙F-distl {zero} {zero} {one} = refl
⊙F-distl {zero} {forth} {zero} = refl
⊙F-distl {zero} {forth} {forth} = refl
⊙F-distl {zero} {forth} {half} = refl
⊙F-distl {zero} {forth} {one} = refl
⊙F-distl {zero} {half} {zero} = refl
⊙F-distl {zero} {half} {forth} = refl
⊙F-distl {zero} {half} {half} = refl
⊙F-distl {zero} {half} {one} = refl
⊙F-distl {zero} {one} {zero} = refl
⊙F-distl {zero} {one} {forth} = refl
⊙F-distl {zero} {one} {half} = refl
⊙F-distl {zero} {one} {one} = refl
⊙F-distl {forth} {zero} {zero} = refl
⊙F-distl {forth} {zero} {forth} = refl
⊙F-distl {forth} {zero} {half} = refl
⊙F-distl {forth} {zero} {one} = refl
⊙F-distl {forth} {forth} {zero} = refl
⊙F-distl {forth} {forth} {forth} = refl
⊙F-distl {forth} {forth} {half} = refl
⊙F-distl {forth} {forth} {one} = refl
⊙F-distl {forth} {half} {zero} = refl
⊙F-distl {forth} {half} {forth} = refl
⊙F-distl {forth} {half} {half} = refl
⊙F-distl {forth} {half} {one} = refl
⊙F-distl {forth} {one} {zero} = refl
⊙F-distl {forth} {one} {forth} = refl
⊙F-distl {forth} {one} {half} = refl
⊙F-distl {forth} {one} {one} = refl
⊙F-distl {half} {zero} {zero} = refl
⊙F-distl {half} {zero} {forth} = refl
⊙F-distl {half} {zero} {half} = refl
⊙F-distl {half} {zero} {one} = refl
⊙F-distl {half} {forth} {zero} = refl
⊙F-distl {half} {forth} {forth} = refl
⊙F-distl {half} {forth} {half} = refl
⊙F-distl {half} {forth} {one} = refl
⊙F-distl {half} {half} {zero} = refl
⊙F-distl {half} {half} {forth} = refl
⊙F-distl {half} {half} {half} = refl
⊙F-distl {half} {half} {one} = refl
⊙F-distl {half} {one} {zero} = refl
⊙F-distl {half} {one} {forth} = refl
⊙F-distl {half} {one} {half} = refl
⊙F-distl {half} {one} {one} = refl
⊙F-distl {one} {zero} {zero} = refl
⊙F-distl {one} {zero} {forth} = refl
⊙F-distl {one} {zero} {half} = refl
⊙F-distl {one} {zero} {one} = refl
⊙F-distl {one} {forth} {zero} = refl
⊙F-distl {one} {forth} {forth} = refl
⊙F-distl {one} {forth} {half} = refl
⊙F-distl {one} {forth} {one} = refl
⊙F-distl {one} {half} {zero} = refl
⊙F-distl {one} {half} {forth} = refl
⊙F-distl {one} {half} {half} = refl
⊙F-distl {one} {half} {one} = refl
⊙F-distl {one} {one} {zero} = refl
⊙F-distl {one} {one} {forth} = refl
⊙F-distl {one} {one} {half} = refl
⊙F-distl {one} {one} {one} = refl
⊙F-distr : {a b c : Four} → ((a ⊔F b) ⊙F c) ≡ ((a ⊙F c) ⊔F (b ⊙F c))
⊙F-distr {zero} {zero} {zero} = refl
⊙F-distr {zero} {zero} {forth} = refl
⊙F-distr {zero} {zero} {half} = refl
⊙F-distr {zero} {zero} {one} = refl
⊙F-distr {zero} {forth} {zero} = refl
⊙F-distr {zero} {forth} {forth} = refl
⊙F-distr {zero} {forth} {half} = refl
⊙F-distr {zero} {forth} {one} = refl
⊙F-distr {zero} {half} {zero} = refl
⊙F-distr {zero} {half} {forth} = refl
⊙F-distr {zero} {half} {half} = refl
⊙F-distr {zero} {half} {one} = refl
⊙F-distr {zero} {one} {zero} = refl
⊙F-distr {zero} {one} {forth} = refl
⊙F-distr {zero} {one} {half} = refl
⊙F-distr {zero} {one} {one} = refl
⊙F-distr {forth} {zero} {zero} = refl
⊙F-distr {forth} {zero} {forth} = refl
⊙F-distr {forth} {zero} {half} = refl
⊙F-distr {forth} {zero} {one} = refl
⊙F-distr {forth} {forth} {zero} = refl
⊙F-distr {forth} {forth} {forth} = refl
⊙F-distr {forth} {forth} {half} = refl
⊙F-distr {forth} {forth} {one} = refl
⊙F-distr {forth} {half} {zero} = refl
⊙F-distr {forth} {half} {forth} = refl
⊙F-distr {forth} {half} {half} = refl
⊙F-distr {forth} {half} {one} = refl
⊙F-distr {forth} {one} {zero} = refl
⊙F-distr {forth} {one} {forth} = refl
⊙F-distr {forth} {one} {half} = refl
⊙F-distr {forth} {one} {one} = refl
⊙F-distr {half} {zero} {zero} = refl
⊙F-distr {half} {zero} {forth} = refl
⊙F-distr {half} {zero} {half} = refl
⊙F-distr {half} {zero} {one} = refl
⊙F-distr {half} {forth} {zero} = refl
⊙F-distr {half} {forth} {forth} = refl
⊙F-distr {half} {forth} {half} = refl
⊙F-distr {half} {forth} {one} = refl
⊙F-distr {half} {half} {zero} = refl
⊙F-distr {half} {half} {forth} = refl
⊙F-distr {half} {half} {half} = refl
⊙F-distr {half} {half} {one} = refl
⊙F-distr {half} {one} {zero} = refl
⊙F-distr {half} {one} {forth} = refl
⊙F-distr {half} {one} {half} = refl
⊙F-distr {half} {one} {one} = refl
⊙F-distr {one} {zero} {zero} = refl
⊙F-distr {one} {zero} {forth} = refl
⊙F-distr {one} {zero} {half} = refl
⊙F-distr {one} {zero} {one} = refl
⊙F-distr {one} {forth} {zero} = refl
⊙F-distr {one} {forth} {forth} = refl
⊙F-distr {one} {forth} {half} = refl
⊙F-distr {one} {forth} {one} = refl
⊙F-distr {one} {half} {zero} = refl
⊙F-distr {one} {half} {forth} = refl
⊙F-distr {one} {half} {half} = refl
⊙F-distr {one} {half} {one} = refl
⊙F-distr {one} {one} {zero} = refl
⊙F-distr {one} {one} {forth} = refl
⊙F-distr {one} {one} {half} = refl
⊙F-distr {one} {one} {one} = refl
▷F-sym : (∀{a b} → (a ▷F b) ≡ (b ▷F a)) → ⊥ {lzero}
▷F-sym p with p {forth}{one}
... | ()
-- ▷F-contract : (∀{a} → (a ▷F a) ≡ a) → ⊥ {lzero}
-- ▷F-contract p with p {half}
-- ... | ()
▷F-zerol : ∀{x} → (zero ▷F x) ≤₄ zero
▷F-zerol {zero} = triv
▷F-zerol {forth} = triv
▷F-zerol {half} = triv
▷F-zerol {one} = triv
▷F-zeror : ∀{x} → (x ▷F zero) ≤₄ zero
▷F-zeror {zero} = triv
▷F-zeror {forth} = triv
▷F-zeror {half} = triv
▷F-zeror {one} = triv
▷F-assoc : {a b c : Four} → a ▷F (b ▷F c) ≡ (a ▷F b) ▷F c
▷F-assoc {zero} {zero} {zero} = refl
▷F-assoc {zero} {zero} {forth} = refl
▷F-assoc {zero} {zero} {half} = refl
▷F-assoc {zero} {zero} {one} = refl
▷F-assoc {zero} {forth} {zero} = refl
▷F-assoc {zero} {forth} {forth} = refl
▷F-assoc {zero} {forth} {half} = refl
▷F-assoc {zero} {forth} {one} = refl
▷F-assoc {zero} {half} {zero} = refl
▷F-assoc {zero} {half} {forth} = refl
▷F-assoc {zero} {half} {half} = refl
▷F-assoc {zero} {half} {one} = refl
▷F-assoc {zero} {one} {zero} = refl
▷F-assoc {zero} {one} {forth} = refl
▷F-assoc {zero} {one} {half} = refl
▷F-assoc {zero} {one} {one} = refl
▷F-assoc {forth} {zero} {zero} = refl
▷F-assoc {forth} {zero} {forth} = refl
▷F-assoc {forth} {zero} {half} = refl
▷F-assoc {forth} {zero} {one} = refl
▷F-assoc {forth} {forth} {zero} = refl
▷F-assoc {forth} {forth} {forth} = refl
▷F-assoc {forth} {forth} {half} = refl
▷F-assoc {forth} {forth} {one} = refl
▷F-assoc {forth} {half} {zero} = refl
▷F-assoc {forth} {half} {forth} = refl
▷F-assoc {forth} {half} {half} = refl
▷F-assoc {forth} {half} {one} = refl
▷F-assoc {forth} {one} {zero} = refl
▷F-assoc {forth} {one} {forth} = refl
▷F-assoc {forth} {one} {half} = refl
▷F-assoc {forth} {one} {one} = refl
▷F-assoc {half} {zero} {zero} = refl
▷F-assoc {half} {zero} {forth} = refl
▷F-assoc {half} {zero} {half} = refl
▷F-assoc {half} {zero} {one} = refl
▷F-assoc {half} {forth} {zero} = refl
▷F-assoc {half} {forth} {forth} = refl
▷F-assoc {half} {forth} {half} = refl
▷F-assoc {half} {forth} {one} = refl
▷F-assoc {half} {half} {zero} = refl
▷F-assoc {half} {half} {forth} = refl
▷F-assoc {half} {half} {half} = refl
▷F-assoc {half} {half} {one} = refl
▷F-assoc {half} {one} {zero} = refl
▷F-assoc {half} {one} {forth} = refl
▷F-assoc {half} {one} {half} = refl
▷F-assoc {half} {one} {one} = refl
▷F-assoc {one} {zero} {zero} = refl
▷F-assoc {one} {zero} {forth} = refl
▷F-assoc {one} {zero} {half} = refl
▷F-assoc {one} {zero} {one} = refl
▷F-assoc {one} {forth} {zero} = refl
▷F-assoc {one} {forth} {forth} = refl
▷F-assoc {one} {forth} {half} = refl
▷F-assoc {one} {forth} {one} = refl
▷F-assoc {one} {half} {zero} = refl
▷F-assoc {one} {half} {forth} = refl
▷F-assoc {one} {half} {half} = refl
▷F-assoc {one} {half} {one} = refl
▷F-assoc {one} {one} {zero} = refl
▷F-assoc {one} {one} {forth} = refl
▷F-assoc {one} {one} {half} = refl
▷F-assoc {one} {one} {one} = refl
▷F-func : {a c b d : Four} → a ≤₄ c → b ≤₄ d → (a ▷F b) ≤₄ (c ▷F d)
▷F-func {zero} {zero} {zero} {zero} p₁ p₂ = triv
▷F-func {zero} {zero} {zero} {forth} p₁ p₂ = triv
▷F-func {zero} {zero} {zero} {half} p₁ p₂ = triv
▷F-func {zero} {zero} {zero} {one} p₁ p₂ = triv
▷F-func {zero} {zero} {forth} {zero} p₁ p₂ = triv
▷F-func {zero} {zero} {forth} {forth} p₁ p₂ = triv
▷F-func {zero} {zero} {forth} {half} p₁ p₂ = triv
▷F-func {zero} {zero} {forth} {one} p₁ p₂ = triv
▷F-func {zero} {zero} {half} {zero} p₁ p₂ = triv
▷F-func {zero} {zero} {half} {forth} p₁ p₂ = triv
▷F-func {zero} {zero} {half} {half} p₁ p₂ = triv
▷F-func {zero} {zero} {half} {one} p₁ p₂ = triv
▷F-func {zero} {zero} {one} {zero} p₁ p₂ = triv
▷F-func {zero} {zero} {one} {forth} p₁ p₂ = triv
▷F-func {zero} {zero} {one} {half} p₁ p₂ = triv
▷F-func {zero} {zero} {one} {one} p₁ p₂ = triv
▷F-func {zero} {forth} {zero} {zero} p₁ p₂ = triv
▷F-func {zero} {forth} {zero} {forth} p₁ p₂ = triv
▷F-func {zero} {forth} {zero} {half} p₁ p₂ = triv
▷F-func {zero} {forth} {zero} {one} p₁ p₂ = triv
▷F-func {zero} {forth} {forth} {zero} p₁ p₂ = triv
▷F-func {zero} {forth} {forth} {forth} p₁ p₂ = triv
▷F-func {zero} {forth} {forth} {half} p₁ p₂ = triv
▷F-func {zero} {forth} {forth} {one} p₁ p₂ = triv
▷F-func {zero} {forth} {half} {zero} p₁ p₂ = triv
▷F-func {zero} {forth} {half} {forth} p₁ p₂ = triv
▷F-func {zero} {forth} {half} {half} p₁ p₂ = triv
▷F-func {zero} {forth} {half} {one} p₁ p₂ = triv
▷F-func {zero} {forth} {one} {zero} p₁ p₂ = triv
▷F-func {zero} {forth} {one} {forth} p₁ p₂ = triv
▷F-func {zero} {forth} {one} {half} p₁ p₂ = triv
▷F-func {zero} {forth} {one} {one} p₁ p₂ = triv
▷F-func {zero} {half} {zero} {zero} p₁ p₂ = triv
▷F-func {zero} {half} {zero} {forth} p₁ p₂ = triv
▷F-func {zero} {half} {zero} {half} p₁ p₂ = triv
▷F-func {zero} {half} {zero} {one} p₁ p₂ = triv
▷F-func {zero} {half} {forth} {zero} p₁ p₂ = triv
▷F-func {zero} {half} {forth} {forth} p₁ p₂ = triv
▷F-func {zero} {half} {forth} {half} p₁ p₂ = triv
▷F-func {zero} {half} {forth} {one} p₁ p₂ = triv
▷F-func {zero} {half} {half} {zero} p₁ p₂ = triv
▷F-func {zero} {half} {half} {forth} p₁ p₂ = triv
▷F-func {zero} {half} {half} {half} p₁ p₂ = triv
▷F-func {zero} {half} {half} {one} p₁ p₂ = triv
▷F-func {zero} {half} {one} {zero} p₁ p₂ = triv
▷F-func {zero} {half} {one} {forth} p₁ p₂ = triv
▷F-func {zero} {half} {one} {half} p₁ p₂ = triv
▷F-func {zero} {half} {one} {one} p₁ p₂ = triv
▷F-func {zero} {one} {zero} {zero} p₁ p₂ = triv
▷F-func {zero} {one} {zero} {forth} p₁ p₂ = triv
▷F-func {zero} {one} {zero} {half} p₁ p₂ = triv
▷F-func {zero} {one} {zero} {one} p₁ p₂ = triv
▷F-func {zero} {one} {forth} {zero} p₁ p₂ = triv
▷F-func {zero} {one} {forth} {forth} p₁ p₂ = triv
▷F-func {zero} {one} {forth} {half} p₁ p₂ = triv
▷F-func {zero} {one} {forth} {one} p₁ p₂ = triv
▷F-func {zero} {one} {half} {zero} p₁ p₂ = triv
▷F-func {zero} {one} {half} {forth} p₁ p₂ = triv
▷F-func {zero} {one} {half} {half} p₁ p₂ = triv
▷F-func {zero} {one} {half} {one} p₁ p₂ = triv
▷F-func {zero} {one} {one} {zero} p₁ p₂ = triv
▷F-func {zero} {one} {one} {forth} p₁ p₂ = triv
▷F-func {zero} {one} {one} {half} p₁ p₂ = triv
▷F-func {zero} {one} {one} {one} p₁ p₂ = triv
▷F-func {forth} {zero} {zero} {zero} p₁ p₂ = triv
▷F-func {forth} {zero} {zero} {forth} p₁ p₂ = triv
▷F-func {forth} {zero} {zero} {half} p₁ p₂ = triv
▷F-func {forth} {zero} {zero} {one} p₁ p₂ = triv
▷F-func {forth} {zero} {forth} {zero} () ()
▷F-func {forth} {zero} {forth} {forth} () p₂
▷F-func {forth} {zero} {forth} {half} () p₂
▷F-func {forth} {zero} {forth} {one} () p₂
▷F-func {forth} {zero} {half} {zero} () ()
▷F-func {forth} {zero} {half} {forth} () ()
▷F-func {forth} {zero} {half} {half} () p₂
▷F-func {forth} {zero} {half} {one} () p₂
▷F-func {forth} {zero} {one} {zero} () ()
▷F-func {forth} {zero} {one} {forth} () ()
▷F-func {forth} {zero} {one} {half} () ()
▷F-func {forth} {zero} {one} {one} () p₂
▷F-func {forth} {forth} {zero} {zero} p₁ p₂ = triv
▷F-func {forth} {forth} {zero} {forth} p₁ p₂ = triv
▷F-func {forth} {forth} {zero} {half} p₁ p₂ = triv
▷F-func {forth} {forth} {zero} {one} p₁ p₂ = triv
▷F-func {forth} {forth} {forth} {zero} p₁ ()
▷F-func {forth} {forth} {forth} {forth} p₁ p₂ = triv
▷F-func {forth} {forth} {forth} {half} p₁ p₂ = triv
▷F-func {forth} {forth} {forth} {one} p₁ p₂ = triv
▷F-func {forth} {forth} {half} {zero} p₁ ()
▷F-func {forth} {forth} {half} {forth} p₁ ()
▷F-func {forth} {forth} {half} {half} p₁ p₂ = triv
▷F-func {forth} {forth} {half} {one} p₁ p₂ = triv
▷F-func {forth} {forth} {one} {zero} p₁ ()
▷F-func {forth} {forth} {one} {forth} p₁ ()
▷F-func {forth} {forth} {one} {half} p₁ p₂ = triv
▷F-func {forth} {forth} {one} {one} p₁ p₂ = triv
▷F-func {forth} {half} {zero} {zero} p₁ p₂ = triv
▷F-func {forth} {half} {zero} {forth} p₁ p₂ = triv
▷F-func {forth} {half} {zero} {half} p₁ p₂ = triv
▷F-func {forth} {half} {zero} {one} p₁ p₂ = triv
▷F-func {forth} {half} {forth} {zero} p₁ ()
▷F-func {forth} {half} {forth} {forth} p₁ p₂ = triv
▷F-func {forth} {half} {forth} {half} p₁ p₂ = triv
▷F-func {forth} {half} {forth} {one} p₁ p₂ = triv
▷F-func {forth} {half} {half} {zero} p₁ ()
▷F-func {forth} {half} {half} {forth} p₁ p₂ = triv
▷F-func {forth} {half} {half} {half} p₁ p₂ = triv
▷F-func {forth} {half} {half} {one} p₁ p₂ = triv
▷F-func {forth} {half} {one} {zero} p₁ ()
▷F-func {forth} {half} {one} {forth} p₁ p₂ = triv
▷F-func {forth} {half} {one} {half} p₁ p₂ = triv
▷F-func {forth} {half} {one} {one} p₁ p₂ = triv
▷F-func {forth} {one} {zero} {zero} p₁ p₂ = triv
▷F-func {forth} {one} {zero} {forth} p₁ p₂ = triv
▷F-func {forth} {one} {zero} {half} p₁ p₂ = triv
▷F-func {forth} {one} {zero} {one} p₁ p₂ = triv
▷F-func {forth} {one} {forth} {zero} p₁ ()
▷F-func {forth} {one} {forth} {forth} p₁ p₂ = triv
▷F-func {forth} {one} {forth} {half} p₁ p₂ = triv
▷F-func {forth} {one} {forth} {one} p₁ p₂ = triv
▷F-func {forth} {one} {half} {zero} p₁ ()
▷F-func {forth} {one} {half} {forth} p₁ p₂ = triv
▷F-func {forth} {one} {half} {half} p₁ p₂ = triv
▷F-func {forth} {one} {half} {one} p₁ p₂ = triv
▷F-func {forth} {one} {one} {zero} p₁ ()
▷F-func {forth} {one} {one} {forth} p₁ p₂ = triv
▷F-func {forth} {one} {one} {half} p₁ p₂ = triv
▷F-func {forth} {one} {one} {one} p₁ p₂ = triv
▷F-func {half} {zero} {zero} {zero} p₁ p₂ = triv
▷F-func {half} {zero} {zero} {forth} p₁ p₂ = triv
▷F-func {half} {zero} {zero} {half} p₁ p₂ = triv
▷F-func {half} {zero} {zero} {one} p₁ p₂ = triv
▷F-func {half} {zero} {forth} {zero} () ()
▷F-func {half} {zero} {forth} {forth} () p₂
▷F-func {half} {zero} {forth} {half} () p₂
▷F-func {half} {zero} {forth} {one} () p₂
▷F-func {half} {zero} {half} {zero} () ()
▷F-func {half} {zero} {half} {forth} () ()
▷F-func {half} {zero} {half} {half} () p₂
▷F-func {half} {zero} {half} {one} () p₂
▷F-func {half} {zero} {one} {zero} () ()
▷F-func {half} {zero} {one} {forth} () ()
▷F-func {half} {zero} {one} {half} () ()
▷F-func {half} {zero} {one} {one} () p₂
▷F-func {half} {forth} {zero} {zero} p₁ p₂ = triv
▷F-func {half} {forth} {zero} {forth} p₁ p₂ = triv
▷F-func {half} {forth} {zero} {half} p₁ p₂ = triv
▷F-func {half} {forth} {zero} {one} p₁ p₂ = triv
▷F-func {half} {forth} {forth} {zero} () ()
▷F-func {half} {forth} {forth} {forth} () p₂
▷F-func {half} {forth} {forth} {half} () p₂
▷F-func {half} {forth} {forth} {one} () p₂
▷F-func {half} {forth} {half} {zero} () ()
▷F-func {half} {forth} {half} {forth} () ()
▷F-func {half} {forth} {half} {half} () p₂
▷F-func {half} {forth} {half} {one} () p₂
▷F-func {half} {forth} {one} {zero} () ()
▷F-func {half} {forth} {one} {forth} () ()
▷F-func {half} {forth} {one} {half} () ()
▷F-func {half} {forth} {one} {one} () p₂
▷F-func {half} {half} {zero} {zero} p₁ p₂ = triv
▷F-func {half} {half} {zero} {forth} p₁ p₂ = triv
▷F-func {half} {half} {zero} {half} p₁ p₂ = triv
▷F-func {half} {half} {zero} {one} p₁ p₂ = triv
▷F-func {half} {half} {forth} {zero} p₁ ()
▷F-func {half} {half} {forth} {forth} p₁ p₂ = triv
▷F-func {half} {half} {forth} {half} p₁ p₂ = triv
▷F-func {half} {half} {forth} {one} p₁ p₂ = triv
▷F-func {half} {half} {half} {zero} p₁ ()
▷F-func {half} {half} {half} {forth} p₁ p₂ = triv
▷F-func {half} {half} {half} {half} p₁ p₂ = triv
▷F-func {half} {half} {half} {one} p₁ p₂ = triv
▷F-func {half} {half} {one} {zero} p₁ ()
▷F-func {half} {half} {one} {forth} p₁ p₂ = triv
▷F-func {half} {half} {one} {half} p₁ p₂ = triv
▷F-func {half} {half} {one} {one} p₁ p₂ = triv
▷F-func {half} {one} {zero} {zero} p₁ p₂ = triv
▷F-func {half} {one} {zero} {forth} p₁ p₂ = triv
▷F-func {half} {one} {zero} {half} p₁ p₂ = triv
▷F-func {half} {one} {zero} {one} p₁ p₂ = triv
▷F-func {half} {one} {forth} {zero} p₁ ()
▷F-func {half} {one} {forth} {forth} p₁ p₂ = triv
▷F-func {half} {one} {forth} {half} p₁ p₂ = triv
▷F-func {half} {one} {forth} {one} p₁ p₂ = triv
▷F-func {half} {one} {half} {zero} p₁ ()
▷F-func {half} {one} {half} {forth} p₁ p₂ = triv
▷F-func {half} {one} {half} {half} p₁ p₂ = triv
▷F-func {half} {one} {half} {one} p₁ p₂ = triv
▷F-func {half} {one} {one} {zero} p₁ ()
▷F-func {half} {one} {one} {forth} p₁ p₂ = triv
▷F-func {half} {one} {one} {half} p₁ p₂ = triv
▷F-func {half} {one} {one} {one} p₁ p₂ = triv
▷F-func {one} {zero} {zero} {zero} p₁ p₂ = triv
▷F-func {one} {zero} {zero} {forth} p₁ p₂ = triv
▷F-func {one} {zero} {zero} {half} p₁ p₂ = triv
▷F-func {one} {zero} {zero} {one} p₁ p₂ = triv
▷F-func {one} {zero} {forth} {zero} p₁ ()
▷F-func {one} {zero} {forth} {forth} () p₂
▷F-func {one} {zero} {forth} {half} () p₂
▷F-func {one} {zero} {forth} {one} () p₂
▷F-func {one} {zero} {half} {zero} () ()
▷F-func {one} {zero} {half} {forth} () ()
▷F-func {one} {zero} {half} {half} () p₂
▷F-func {one} {zero} {half} {one} () p₂
▷F-func {one} {zero} {one} {zero} () ()
▷F-func {one} {zero} {one} {forth} () ()
▷F-func {one} {zero} {one} {half} () ()
▷F-func {one} {zero} {one} {one} () p₂
▷F-func {one} {forth} {zero} {zero} p₁ p₂ = triv
▷F-func {one} {forth} {zero} {forth} p₁ p₂ = triv
▷F-func {one} {forth} {zero} {half} p₁ p₂ = triv
▷F-func {one} {forth} {zero} {one} p₁ p₂ = triv
▷F-func {one} {forth} {forth} {zero} () ()
▷F-func {one} {forth} {forth} {forth} () p₂
▷F-func {one} {forth} {forth} {half} () p₂
▷F-func {one} {forth} {forth} {one} () p₂
▷F-func {one} {forth} {half} {zero} () ()
▷F-func {one} {forth} {half} {forth} () ()
▷F-func {one} {forth} {half} {half} () p₂
▷F-func {one} {forth} {half} {one} () p₂
▷F-func {one} {forth} {one} {zero} () ()
▷F-func {one} {forth} {one} {forth} () ()
▷F-func {one} {forth} {one} {half} () ()
▷F-func {one} {forth} {one} {one} () p₂
▷F-func {one} {half} {zero} {zero} p₁ p₂ = triv
▷F-func {one} {half} {zero} {forth} p₁ p₂ = triv
▷F-func {one} {half} {zero} {half} p₁ p₂ = triv
▷F-func {one} {half} {zero} {one} p₁ p₂ = triv
▷F-func {one} {half} {forth} {zero} () ()
▷F-func {one} {half} {forth} {forth} () p₂
▷F-func {one} {half} {forth} {half} () p₂
▷F-func {one} {half} {forth} {one} () p₂
▷F-func {one} {half} {half} {zero} () ()
▷F-func {one} {half} {half} {forth} p₁ p₂ = triv
▷F-func {one} {half} {half} {half} p₁ p₂ = triv
▷F-func {one} {half} {half} {one} p₁ p₂ = triv
▷F-func {one} {half} {one} {zero} () ()
▷F-func {one} {half} {one} {forth} p₁ p₂ = triv
▷F-func {one} {half} {one} {half} p₁ p₂ = triv
▷F-func {one} {half} {one} {one} p₁ p₂ = triv
▷F-func {one} {one} {zero} {zero} p₁ p₂ = triv
▷F-func {one} {one} {zero} {forth} p₁ p₂ = triv
▷F-func {one} {one} {zero} {half} p₁ p₂ = triv
▷F-func {one} {one} {zero} {one} p₁ p₂ = triv
▷F-func {one} {one} {forth} {zero} p₁ ()
▷F-func {one} {one} {forth} {forth} p₁ p₂ = triv
▷F-func {one} {one} {forth} {half} p₁ p₂ = triv
▷F-func {one} {one} {forth} {one} p₁ p₂ = triv
▷F-func {one} {one} {half} {zero} p₁ ()
▷F-func {one} {one} {half} {forth} p₁ p₂ = triv
▷F-func {one} {one} {half} {half} p₁ p₂ = triv
▷F-func {one} {one} {half} {one} p₁ p₂ = triv
▷F-func {one} {one} {one} {zero} p₁ ()
▷F-func {one} {one} {one} {forth} p₁ p₂ = triv
▷F-func {one} {one} {one} {half} p₁ p₂ = triv
▷F-func {one} {one} {one} {one} p₁ p₂ = triv
▷F-distl : {a b c : Four} → (a ▷F (b ⊔F c)) ≡ ((a ▷F b) ⊔F (a ▷F c))
▷F-distl {zero} {zero} {zero} = refl
▷F-distl {zero} {zero} {forth} = refl
▷F-distl {zero} {zero} {half} = refl
▷F-distl {zero} {zero} {one} = refl
▷F-distl {zero} {forth} {zero} = refl
▷F-distl {zero} {forth} {forth} = refl
▷F-distl {zero} {forth} {half} = refl
▷F-distl {zero} {forth} {one} = refl
▷F-distl {zero} {half} {zero} = refl
▷F-distl {zero} {half} {forth} = refl
▷F-distl {zero} {half} {half} = refl
▷F-distl {zero} {half} {one} = refl
▷F-distl {zero} {one} {zero} = refl
▷F-distl {zero} {one} {forth} = refl
▷F-distl {zero} {one} {half} = refl
▷F-distl {zero} {one} {one} = refl
▷F-distl {forth} {zero} {zero} = refl
▷F-distl {forth} {zero} {forth} = refl
▷F-distl {forth} {zero} {half} = refl
▷F-distl {forth} {zero} {one} = refl
▷F-distl {forth} {forth} {zero} = refl
▷F-distl {forth} {forth} {forth} = refl
▷F-distl {forth} {forth} {half} = refl
▷F-distl {forth} {forth} {one} = refl
▷F-distl {forth} {half} {zero} = refl
▷F-distl {forth} {half} {forth} = refl
▷F-distl {forth} {half} {half} = refl
▷F-distl {forth} {half} {one} = refl
▷F-distl {forth} {one} {zero} = refl
▷F-distl {forth} {one} {forth} = refl
▷F-distl {forth} {one} {half} = refl
▷F-distl {forth} {one} {one} = refl
▷F-distl {half} {zero} {zero} = refl
▷F-distl {half} {zero} {forth} = refl
▷F-distl {half} {zero} {half} = refl
▷F-distl {half} {zero} {one} = refl
▷F-distl {half} {forth} {zero} = refl
▷F-distl {half} {forth} {forth} = refl
▷F-distl {half} {forth} {half} = refl
▷F-distl {half} {forth} {one} = refl
▷F-distl {half} {half} {zero} = refl
▷F-distl {half} {half} {forth} = refl
▷F-distl {half} {half} {half} = refl
▷F-distl {half} {half} {one} = refl
▷F-distl {half} {one} {zero} = refl
▷F-distl {half} {one} {forth} = refl
▷F-distl {half} {one} {half} = refl
▷F-distl {half} {one} {one} = refl
▷F-distl {one} {zero} {zero} = refl
▷F-distl {one} {zero} {forth} = refl
▷F-distl {one} {zero} {half} = refl
▷F-distl {one} {zero} {one} = refl
▷F-distl {one} {forth} {zero} = refl
▷F-distl {one} {forth} {forth} = refl
▷F-distl {one} {forth} {half} = refl
▷F-distl {one} {forth} {one} = refl
▷F-distl {one} {half} {zero} = refl
▷F-distl {one} {half} {forth} = refl
▷F-distl {one} {half} {half} = refl
▷F-distl {one} {half} {one} = refl
▷F-distl {one} {one} {zero} = refl
▷F-distl {one} {one} {forth} = refl
▷F-distl {one} {one} {half} = refl
▷F-distl {one} {one} {one} = refl
▷F-distr : {a b c : Four} → ((a ⊔F b) ▷F c) ≡ ((a ▷F c) ⊔F (b ▷F c))
▷F-distr {zero} {zero} {zero} = refl
▷F-distr {zero} {zero} {forth} = refl
▷F-distr {zero} {zero} {half} = refl
▷F-distr {zero} {zero} {one} = refl
▷F-distr {zero} {forth} {zero} = refl
▷F-distr {zero} {forth} {forth} = refl
▷F-distr {zero} {forth} {half} = refl
▷F-distr {zero} {forth} {one} = refl
▷F-distr {zero} {half} {zero} = refl
▷F-distr {zero} {half} {forth} = refl
▷F-distr {zero} {half} {half} = refl
▷F-distr {zero} {half} {one} = refl
▷F-distr {zero} {one} {zero} = refl
▷F-distr {zero} {one} {forth} = refl
▷F-distr {zero} {one} {half} = refl
▷F-distr {zero} {one} {one} = refl
▷F-distr {forth} {zero} {zero} = refl
▷F-distr {forth} {zero} {forth} = refl
▷F-distr {forth} {zero} {half} = refl
▷F-distr {forth} {zero} {one} = refl
▷F-distr {forth} {forth} {zero} = refl
▷F-distr {forth} {forth} {forth} = refl
▷F-distr {forth} {forth} {half} = refl
▷F-distr {forth} {forth} {one} = refl
▷F-distr {forth} {half} {zero} = refl
▷F-distr {forth} {half} {forth} = refl
▷F-distr {forth} {half} {half} = refl
▷F-distr {forth} {half} {one} = refl
▷F-distr {forth} {one} {zero} = refl
▷F-distr {forth} {one} {forth} = refl
▷F-distr {forth} {one} {half} = refl
▷F-distr {forth} {one} {one} = refl
▷F-distr {half} {zero} {zero} = refl
▷F-distr {half} {zero} {forth} = refl
▷F-distr {half} {zero} {half} = refl
▷F-distr {half} {zero} {one} = refl
▷F-distr {half} {forth} {zero} = refl
▷F-distr {half} {forth} {forth} = refl
▷F-distr {half} {forth} {half} = refl
▷F-distr {half} {forth} {one} = refl
▷F-distr {half} {half} {zero} = refl
▷F-distr {half} {half} {forth} = refl
▷F-distr {half} {half} {half} = refl
▷F-distr {half} {half} {one} = refl
▷F-distr {half} {one} {zero} = refl
▷F-distr {half} {one} {forth} = refl
▷F-distr {half} {one} {half} = refl
▷F-distr {half} {one} {one} = refl
▷F-distr {one} {zero} {zero} = refl
▷F-distr {one} {zero} {forth} = refl
▷F-distr {one} {zero} {half} = refl
▷F-distr {one} {zero} {one} = refl
▷F-distr {one} {forth} {zero} = refl
▷F-distr {one} {forth} {forth} = refl
▷F-distr {one} {forth} {half} = refl
▷F-distr {one} {forth} {one} = refl
▷F-distr {one} {half} {zero} = refl
▷F-distr {one} {half} {forth} = refl
▷F-distr {one} {half} {half} = refl
▷F-distr {one} {half} {one} = refl
▷F-distr {one} {one} {zero} = refl
▷F-distr {one} {one} {forth} = refl
▷F-distr {one} {one} {half} = refl
▷F-distr {one} {one} {one} = refl
⊔F-sym : ∀{a b} → (a ⊔F b) ≡ (b ⊔F a)
⊔F-sym {zero} {zero} = refl
⊔F-sym {zero} {forth} = refl
⊔F-sym {zero} {half} = refl
⊔F-sym {zero} {one} = refl
⊔F-sym {forth} {zero} = refl
⊔F-sym {forth} {forth} = refl
⊔F-sym {forth} {half} = refl
⊔F-sym {forth} {one} = refl
⊔F-sym {half} {zero} = refl
⊔F-sym {half} {forth} = refl
⊔F-sym {half} {half} = refl
⊔F-sym {half} {one} = refl
⊔F-sym {one} {zero} = refl
⊔F-sym {one} {forth} = refl
⊔F-sym {one} {half} = refl
⊔F-sym {one} {one} = refl
⊔F-assoc : ∀{a b c} → (a ⊔F b) ⊔F c ≡ a ⊔F (b ⊔F c)
⊔F-assoc {zero} {zero} {zero} = refl
⊔F-assoc {zero} {zero} {forth} = refl
⊔F-assoc {zero} {zero} {half} = refl
⊔F-assoc {zero} {zero} {one} = refl
⊔F-assoc {zero} {forth} {zero} = refl
⊔F-assoc {zero} {forth} {forth} = refl
⊔F-assoc {zero} {forth} {half} = refl