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FLRP.Certificates.SmallLatticeReps.SLR09

A machine-checked representation: Con(B9) ≅ manuscript L9

This is the FLRP.Certificates.SmallLatticeReps.SLR09 module of the Agda Universal Algebra Library.

This module was emitted by scripts/python/flrp/emit_agda.py from scripts/python/flrp/inputs/slr/slr09.json. Do not edit it by hand; rerun the emitter instead.

B9/L9 in the numbering of the DeMeo–Freese–Jipsen manuscript Representing Finite Lattices as Congruence Lattices of Finite Algebras (§ 6 of the 2016-06-10 draft, docs/papers/fin-lat-rep/SmallLatticeReps.tex). The dictionary between the manuscript's lattice numbering and this library's names is docs/notes/flrp-slr-naming.md.

It re-verifies, end-to-end through the WP-6 certificate pipeline (#457), the claim that the congruence lattice of the algebra "B9" — carrier size 16, unary operations f, g, h — is isomorphic to the lattice "manuscript L9" (7 elements). The engine's output below (normal-form parent vectors, Freese traces, and pointer tables) is certificate data only: the search-free checkers of Setoid.Congruences.Certificates re-verify all of it during type-checking, and the FLRP.Certificates assembly turns the checked certificate into the headline theorems — a Representableᵈ witness for the target lattice and a FiniteCongruencesᵈ instance for the algebra. Nothing is believed on the engine's authority: a wrong table or trace would make a decidable check compute to no and break compilation.

{-# OPTIONS --cubical-compatible --exact-split --safe #-}

module FLRP.Certificates.SmallLatticeReps.SLR09 where

-- Imports from Agda and the Agda Standard Library -----------------------------
open import Data.Fin.Base       using ( Fin ; suc )
open import Data.Fin.Patterns   using ( 0F ; 1F ; 2F ; 3F ; 4F ; 5F ; 6F ; 7F ; 8F ; 9F )
open import Data.List.Base      using ( [] ; _∷_ )
open import Data.Vec.Base       using ( Vec ; [] ; _∷_ )
open import Level               using ( 0ℓ )

-- Imports from the Agda Universal Algebra Library -----------------------------
open import Classical.Signatures.Unary   using ( Sig-Unary )
open import Classical.Structures.Unary   using ( tablesToUnaryAlgebra
                                               ; tablesToUnaryAlgebra-FiniteAlgebra
                                               ; Sig-Unary-Fin-FiniteSignature )
open import FLRP.Certificates            using ( MeetMatches ; meetMatches?
                                               ; certRepresentableᵈ )
open import FLRP.Problem                 using ( FiniteLattice ; toLattice )
open import FLRP.Representable           using ( Representableᵈ )
open import Overture.Cayley              using ( Table ; ⟦_⟧ ; from-yes )
open import Overture.Operations.Properties
                                         using ( Associative? ; Commutative?
                                               ; Idempotent? ; Absorbsˡ? ; Absorbsʳ? )
open import Setoid.Algebras.Basic        using ( Algebra )
open import Setoid.Algebras.Finite       using ( FiniteAlgebra )
open import Setoid.Congruences.Certificates.Schema
                                         using ( ParentVec ; Trace ; LatticeCert
                                               ; mkLatticeCert ; mkMerge
                                               ; seed ; translate )
open import Setoid.Congruences.Certificates.Congruence
                                         using ( module CertCheck )
open import Setoid.Congruences.Certificates.Lattice
                                         using ( module LatticeCheck )
open import Setoid.Congruences.Finite.Decidable
                                         using ( FiniteCongruencesᵈ )
open import Setoid.Signatures.Finite     using ( FiniteSignature )

-- Data.Fin.Patterns stops at 9F; the larger literals this
-- module needs are pattern synonyms in the same style.
pattern 10F = suc 9F
pattern 11F = suc 10F
pattern 12F = suc 11F
pattern 13F = suc 12F
pattern 14F = suc 13F
pattern 15F = suc 14F

The algebra, from its operation tables

Row f of opTables is the value table of the f-th unary operation; Classical.Structures.Unary turns the table into the algebra and its finiteness witnesses.

opTables : Vec (Vec (Fin 16) 16) 3
opTables = (0F  0F  0F  0F  0F  0F  2F  1F  2F  1F  3F  4F  5F  3F  4F  5F  [])
          (0F  0F  0F  0F  0F  0F  6F  7F  6F  7F  10F  11F  12F  10F  11F  12F  [])
          (13F  14F  15F  1F  9F  8F  15F  14F  13F  15F  1F  9F  8F  8F  1F  9F  [])
          []

𝑨 : Algebra {𝑆 = Sig-Unary (Fin 3)} 0ℓ 0ℓ
𝑨 = tablesToUnaryAlgebra 16 3 opTables

𝑭 : FiniteAlgebra 𝑨
𝑭 = tablesToUnaryAlgebra-FiniteAlgebra 16 3 opTables

𝑺 : FiniteSignature (Sig-Unary (Fin 3))
𝑺 = Sig-Unary-Fin-FiniteSignature 3

The target lattice, from its Cayley tables

The claimed congruence lattice "manuscript L9", presented exactly as the worked lattice examples are (Examples.Classical.Lattices.L7): meet and join tables, every law discharged by decision over the finite carrier.

∧-table ∨-table : Table 7
∧-table = (0F  0F  0F  0F  0F  0F  0F  [])
         (0F  1F  0F  0F  0F  1F  1F  [])
         (0F  0F  2F  0F  0F  0F  2F  [])
         (0F  0F  0F  3F  0F  0F  3F  [])
         (0F  0F  0F  0F  4F  0F  4F  [])
         (0F  1F  0F  0F  0F  5F  5F  [])
         (0F  1F  2F  3F  4F  5F  6F  [])
         []

∨-table = (0F  1F  2F  3F  4F  5F  6F  [])
         (1F  1F  6F  6F  6F  5F  6F  [])
         (2F  6F  2F  6F  6F  6F  6F  [])
         (3F  6F  6F  3F  6F  6F  6F  [])
         (4F  6F  6F  6F  4F  6F  6F  [])
         (5F  5F  6F  6F  6F  5F  6F  [])
         (6F  6F  6F  6F  6F  6F  6F  [])
         []

open FiniteLattice

𝑳 : FiniteLattice
𝑳 .size     = 6
𝑳 ._∧_      =  ∧-table 
𝑳 ._∨_      =  ∨-table 
𝑳 .∧-assoc  = from-yes (Associative?  ∧-table )
𝑳 .∧-comm   = from-yes (Commutative?  ∧-table )
𝑳 .∧-idem   = from-yes (Idempotent?  ∧-table )
𝑳 .∨-assoc  = from-yes (Associative?  ∨-table )
𝑳 .∨-comm   = from-yes (Commutative?  ∨-table )
𝑳 .∨-idem   = from-yes (Idempotent?  ∨-table )
𝑳 .absorbˡ  = from-yes (Absorbsˡ?  ∧-table   ∨-table )
𝑳 .absorbʳ  = from-yes (Absorbsʳ?  ∧-table   ∨-table )

The certificate

The engine's whole-lattice certificate (design note § 4): the congruence list as normal-form parent vectors, indexed by the lattice's carrier; the principal-congruence pointer table with one Freese trace per carrier pair; and the join traces, seeded by forest edges. The meet and join tables are the target's own tables, which is what MeetMatches pins.

open CertCheck 𝑭 𝑺 using ( arOf )

cert : LatticeCert 16 3 arOf 7
cert = mkLatticeCert partsᵛ 0F prinᵛ prinTrᵛ ∧-table ∨-table joinTrᵛ
  where
  partsᵛ : Vec (ParentVec 16) 7
  partsᵛ = (0F  1F  2F  3F  4F  5F  6F  7F  8F  9F  10F  11F  12F  13F  14F  15F  [])
          (0F  0F  0F  3F  3F  3F  0F  0F  0F  0F  10F  10F  10F  13F  13F  13F  [])
          (0F  1F  2F  0F  2F  1F  6F  7F  8F  9F  0F  6F  7F  1F  8F  9F  [])
          (0F  1F  2F  1F  0F  2F  6F  7F  8F  9F  7F  0F  6F  9F  1F  8F  [])
          (0F  1F  2F  2F  1F  0F  6F  7F  8F  9F  6F  7F  0F  8F  9F  1F  [])
          (0F  0F  0F  3F  3F  3F  0F  0F  0F  0F  10F  10F  10F  10F  10F  10F  [])
          (0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  0F  [])
          []

  prinᵛ : Vec (Vec (Fin 7) 16) 16
  prinᵛ = (0F  1F  1F  2F  3F  4F  1F  1F  1F  1F  2F  3F  4F  6F  6F  6F  [])
         (1F  0F  1F  3F  4F  2F  1F  1F  1F  1F  6F  6F  6F  2F  3F  4F  [])
         (1F  1F  0F  4F  2F  3F  1F  1F  1F  1F  6F  6F  6F  6F  6F  6F  [])
         (2F  3F  4F  0F  1F  1F  6F  6F  6F  6F  2F  6F  6F  6F  3F  6F  [])
         (3F  4F  2F  1F  0F  1F  6F  6F  6F  6F  6F  3F  6F  6F  6F  4F  [])
         (4F  2F  3F  1F  1F  0F  6F  6F  6F  6F  6F  6F  4F  2F  6F  6F  [])
         (1F  1F  1F  6F  6F  6F  0F  1F  1F  1F  4F  2F  3F  6F  6F  6F  [])
         (1F  1F  1F  6F  6F  6F  1F  0F  1F  1F  3F  4F  2F  6F  6F  6F  [])
         (1F  1F  1F  6F  6F  6F  1F  1F  0F  1F  6F  6F  6F  4F  2F  3F  [])
         (1F  1F  1F  6F  6F  6F  1F  1F  1F  0F  6F  6F  6F  3F  4F  2F  [])
         (2F  6F  6F  2F  6F  6F  4F  3F  6F  6F  0F  1F  1F  5F  5F  5F  [])
         (3F  6F  6F  6F  3F  6F  2F  4F  6F  6F  1F  0F  1F  5F  5F  5F  [])
         (4F  6F  6F  6F  6F  4F  3F  2F  6F  6F  1F  1F  0F  5F  5F  5F  [])
         (6F  2F  6F  6F  6F  2F  6F  6F  4F  3F  5F  5F  5F  0F  1F  1F  [])
         (6F  3F  6F  3F  6F  6F  6F  6F  2F  4F  5F  5F  5F  1F  0F  1F  [])
         (6F  4F  6F  6F  4F  6F  6F  6F  3F  2F  5F  5F  5F  1F  1F  0F  [])
         []

  prinTrᵛ : Vec (Vec (Trace 16 3 arOf) 16) 16
  prinTrᵛ = ( []
             ( mkMerge 0F 1F (seed 0)
               mkMerge 13F 14F (translate 2F 0F (0F  []) 0)
               mkMerge 3F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 0F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 0F 7F (translate 1F 0F (0F  []) 2)
               mkMerge 14F 15F (translate 2F 0F (0F  []) 3)
               mkMerge 4F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 12F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 2F (seed 0)
               mkMerge 13F 15F (translate 2F 0F (0F  []) 0)
               mkMerge 3F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 12F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 1F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 7F (translate 1F 0F (0F  []) 2)
               mkMerge 0F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 14F 13F (translate 2F 0F (0F  []) 3)
               mkMerge 4F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 10F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 3F (seed 0)
               mkMerge 13F 1F (translate 2F 0F (0F  []) 0)
               mkMerge 10F 0F (translate 1F 0F (0F  []) 0)
               mkMerge 8F 14F (translate 2F 0F (0F  []) 1)
               mkMerge 2F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 6F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 15F 9F (translate 2F 0F (0F  []) 1)
               mkMerge 5F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 7F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 4F (seed 0)
               mkMerge 13F 9F (translate 2F 0F (0F  []) 0)
               mkMerge 3F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 5F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 12F (translate 1F 0F (0F  []) 2)
               mkMerge 0F 11F (translate 1F 0F (0F  []) 2)
               [] )
             ( mkMerge 0F 5F (seed 0)
               mkMerge 13F 8F (translate 2F 0F (0F  []) 0)
               mkMerge 3F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 6F (translate 1F 0F (0F  []) 1)
               mkMerge 1F 15F (translate 2F 0F (0F  []) 1)
               mkMerge 0F 12F (translate 1F 0F (0F  []) 0)
               mkMerge 14F 9F (translate 2F 0F (0F  []) 1)
               mkMerge 4F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 7F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 6F (seed 0)
               mkMerge 0F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 13F 15F (translate 2F 0F (0F  []) 1)
               mkMerge 3F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 12F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 1F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 7F (translate 1F 0F (0F  []) 2)
               mkMerge 14F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 4F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 10F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 7F (seed 0)
               mkMerge 0F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 13F 14F (translate 2F 0F (0F  []) 1)
               mkMerge 3F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 0F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 14F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 4F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 12F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 8F (seed 0)
               mkMerge 0F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 0F 6F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 15F (translate 2F 0F (0F  []) 1)
               mkMerge 3F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 12F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 6F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 14F 13F (translate 2F 0F (0F  []) 1)
               mkMerge 4F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 10F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 9F (seed 0)
               mkMerge 0F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 0F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 13F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 5F (translate 0F 0F (0F  []) 1)
               mkMerge 10F 12F (translate 1F 0F (0F  []) 2)
               mkMerge 8F 9F (translate 2F 0F (0F  []) 3)
               mkMerge 3F 4F (translate 0F 0F (0F  []) 3)
               mkMerge 10F 11F (translate 1F 0F (0F  []) 4)
               mkMerge 2F 1F (translate 0F 0F (0F  []) 2)
               mkMerge 6F 7F (translate 1F 0F (0F  []) 3)
               [] )
             ( mkMerge 0F 10F (seed 0)
               mkMerge 0F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 13F 1F (translate 2F 0F (0F  []) 1)
               mkMerge 8F 14F (translate 2F 0F (0F  []) 0)
               mkMerge 2F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 6F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 15F 9F (translate 2F 0F (0F  []) 1)
               mkMerge 5F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 7F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 11F (seed 0)
               mkMerge 0F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 13F 9F (translate 2F 0F (0F  []) 1)
               mkMerge 3F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 5F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 12F (translate 1F 0F (0F  []) 2)
               [] )
             ( mkMerge 0F 12F (seed 0)
               mkMerge 0F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 13F 8F (translate 2F 0F (0F  []) 1)
               mkMerge 3F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 6F (translate 1F 0F (0F  []) 1)
               mkMerge 1F 15F (translate 2F 0F (0F  []) 1)
               mkMerge 14F 9F (translate 2F 0F (0F  []) 0)
               mkMerge 4F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 7F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 0F 13F (seed 0)
               mkMerge 0F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 0F 10F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 13F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 2F (translate 0F 0F (0F  []) 1)
               mkMerge 10F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 8F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 2F 4F (translate 0F 0F (0F  []) 1)
               mkMerge 6F 11F (translate 1F 0F (0F  []) 2)
               mkMerge 0F 5F (translate 0F 0F (0F  []) 2)
               mkMerge 0F 12F (translate 1F 0F (0F  []) 3)
               mkMerge 14F 9F (translate 2F 0F (0F  []) 4)
               mkMerge 11F 7F (translate 1F 0F (0F  []) 0)
               [] )
             ( mkMerge 0F 14F (seed 0)
               mkMerge 0F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 0F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 13F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 0F (translate 0F 0F (0F  []) 1)
               mkMerge 10F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 8F 14F (translate 2F 0F (0F  []) 3)
               mkMerge 3F 1F (translate 0F 0F (0F  []) 3)
               mkMerge 10F 7F (translate 1F 0F (0F  []) 4)
               mkMerge 8F 15F (translate 2F 0F (0F  []) 5)
               mkMerge 2F 4F (translate 0F 0F (0F  []) 3)
               mkMerge 6F 11F (translate 1F 0F (0F  []) 4)
               mkMerge 2F 5F (translate 0F 0F (0F  []) 2)
               mkMerge 6F 12F (translate 1F 0F (0F  []) 3)
               [] )
             ( mkMerge 0F 15F (seed 0)
               mkMerge 0F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 0F 12F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 13F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 1F (translate 0F 0F (0F  []) 1)
               mkMerge 10F 7F (translate 1F 0F (0F  []) 2)
               mkMerge 8F 15F (translate 2F 0F (0F  []) 3)
               mkMerge 3F 2F (translate 0F 0F (0F  []) 3)
               mkMerge 10F 6F (translate 1F 0F (0F  []) 4)
               mkMerge 1F 14F (translate 2F 0F (0F  []) 4)
               mkMerge 2F 5F (translate 0F 0F (0F  []) 3)
               mkMerge 6F 12F (translate 1F 0F (0F  []) 4)
               mkMerge 0F 4F (translate 0F 0F (0F  []) 2)
               mkMerge 0F 11F (translate 1F 0F (0F  []) 3)
               [] )
             [] )
           ( ( mkMerge 1F 0F (seed 0)
               mkMerge 14F 13F (translate 2F 0F (0F  []) 0)
               mkMerge 4F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 10F (translate 1F 0F (0F  []) 1)
               mkMerge 1F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 9F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 0F 2F (translate 0F 0F (0F  []) 1)
               mkMerge 0F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 7F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 15F 14F (translate 2F 0F (0F  []) 3)
               mkMerge 5F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 11F (translate 1F 0F (0F  []) 1)
               [] )
             []
             ( mkMerge 1F 2F (seed 0)
               mkMerge 14F 15F (translate 2F 0F (0F  []) 0)
               mkMerge 4F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 12F (translate 1F 0F (0F  []) 1)
               mkMerge 1F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 9F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 0F 1F (translate 0F 0F (0F  []) 1)
               mkMerge 0F 7F (translate 1F 0F (0F  []) 2)
               mkMerge 7F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 15F 13F (translate 2F 0F (0F  []) 3)
               mkMerge 5F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 10F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 1F 3F (seed 0)
               mkMerge 14F 1F (translate 2F 0F (0F  []) 0)
               mkMerge 4F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 0F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 13F (translate 2F 0F (0F  []) 1)
               mkMerge 7F 10F (translate 1F 0F (0F  []) 0)
               mkMerge 15F 8F (translate 2F 0F (0F  []) 1)
               mkMerge 5F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 6F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 1F 4F (seed 0)
               mkMerge 14F 9F (translate 2F 0F (0F  []) 0)
               mkMerge 11F 7F (translate 1F 0F (0F  []) 0)
               mkMerge 1F 15F (translate 2F 0F (0F  []) 1)
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               mkMerge 8F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 4F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 11F (translate 1F 0F (0F  []) 2)
               mkMerge 2F 3F (translate 0F 0F (0F  []) 2)
               mkMerge 6F 10F (translate 1F 0F (0F  []) 3)
               [] )
             ( mkMerge 15F 2F (seed 0)
               mkMerge 5F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 0F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 5F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 12F (translate 1F 0F (0F  []) 2)
               mkMerge 2F 3F (translate 0F 0F (0F  []) 2)
               mkMerge 6F 10F (translate 1F 0F (0F  []) 3)
               mkMerge 14F 8F (translate 2F 0F (0F  []) 3)
               mkMerge 15F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 4F 2F (translate 0F 0F (0F  []) 1)
               mkMerge 11F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 1F 13F (translate 2F 0F (0F  []) 3)
               mkMerge 0F 10F (translate 1F 0F (0F  []) 0)
               [] )
             ( mkMerge 15F 3F (seed 0)
               mkMerge 5F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 0F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 0F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 15F 14F (translate 2F 0F (0F  []) 3)
               mkMerge 2F 3F (translate 0F 0F (0F  []) 3)
               mkMerge 6F 10F (translate 1F 0F (0F  []) 4)
               mkMerge 14F 13F (translate 2F 0F (0F  []) 4)
               mkMerge 5F 4F (translate 0F 0F (0F  []) 3)
               mkMerge 12F 11F (translate 1F 0F (0F  []) 4)
               mkMerge 15F 1F (translate 2F 0F (0F  []) 4)
               mkMerge 11F 10F (translate 1F 0F (0F  []) 3)
               [] )
             ( mkMerge 15F 4F (seed 0)
               mkMerge 5F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 0F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 13F (translate 2F 0F (0F  []) 1)
               mkMerge 2F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 6F 10F (translate 1F 0F (0F  []) 1)
               mkMerge 15F 1F (translate 2F 0F (0F  []) 1)
               mkMerge 9F 14F (translate 2F 0F (0F  []) 0)
               mkMerge 7F 11F (translate 1F 0F (0F  []) 0)
               [] )
             ( mkMerge 15F 5F (seed 0)
               mkMerge 5F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 0F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 2F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 15F 13F (translate 2F 0F (0F  []) 3)
               mkMerge 2F 3F (translate 0F 0F (0F  []) 3)
               mkMerge 6F 10F (translate 1F 0F (0F  []) 4)
               mkMerge 14F 15F (translate 2F 0F (0F  []) 4)
               mkMerge 5F 3F (translate 0F 0F (0F  []) 3)
               mkMerge 12F 10F (translate 1F 0F (0F  []) 4)
               mkMerge 4F 5F (translate 0F 0F (0F  []) 2)
               mkMerge 11F 12F (translate 1F 0F (0F  []) 3)
               [] )
             ( mkMerge 15F 6F (seed 0)
               mkMerge 5F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 6F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 5F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 12F (translate 1F 0F (0F  []) 2)
               mkMerge 13F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 14F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 1F (translate 0F 0F (0F  []) 1)
               mkMerge 10F 7F (translate 1F 0F (0F  []) 2)
               mkMerge 4F 2F (translate 0F 0F (0F  []) 2)
               mkMerge 11F 6F (translate 1F 0F (0F  []) 3)
               mkMerge 1F 13F (translate 2F 0F (0F  []) 4)
               mkMerge 0F 3F (translate 0F 0F (0F  []) 0)
               [] )
             ( mkMerge 15F 7F (seed 0)
               mkMerge 5F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 4F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 11F (translate 1F 0F (0F  []) 2)
               mkMerge 15F 1F (translate 2F 0F (0F  []) 3)
               mkMerge 2F 4F (translate 0F 0F (0F  []) 3)
               mkMerge 6F 11F (translate 1F 0F (0F  []) 4)
               mkMerge 13F 1F (translate 2F 0F (0F  []) 5)
               mkMerge 5F 0F (translate 0F 0F (0F  []) 3)
               mkMerge 15F 9F (translate 2F 0F (0F  []) 3)
               mkMerge 3F 0F (translate 0F 0F (0F  []) 2)
               mkMerge 10F 0F (translate 1F 0F (0F  []) 3)
               [] )
             ( mkMerge 15F 8F (seed 0)
               mkMerge 5F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 6F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 7F 10F (translate 1F 0F (0F  []) 1)
               mkMerge 14F 1F (translate 2F 0F (0F  []) 1)
               mkMerge 4F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 11F 0F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 15F 9F (seed 0)
               mkMerge 5F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 14F (translate 2F 0F (0F  []) 1)
               mkMerge 2F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 6F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 0F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 15F 10F (seed 0)
               mkMerge 5F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 10F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 0F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 15F 14F (translate 2F 0F (0F  []) 3)
               mkMerge 2F 0F (translate 0F 0F (0F  []) 3)
               mkMerge 6F 0F (translate 1F 0F (0F  []) 4)
               mkMerge 13F 14F (translate 2F 0F (0F  []) 5)
               mkMerge 5F 4F (translate 0F 0F (0F  []) 3)
               mkMerge 12F 11F (translate 1F 0F (0F  []) 4)
               [] )
             ( mkMerge 15F 11F (seed 0)
               mkMerge 5F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 8F 9F (translate 2F 0F (0F  []) 1)
               mkMerge 2F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 6F 7F (translate 1F 0F (0F  []) 1)
               mkMerge 13F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 15F 14F (translate 2F 0F (0F  []) 2)
               mkMerge 3F 5F (translate 0F 0F (0F  []) 1)
               mkMerge 10F 12F (translate 1F 0F (0F  []) 2)
               mkMerge 9F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 0F (translate 0F 0F (0F  []) 0)
               mkMerge 7F 0F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 15F 12F (seed 0)
               mkMerge 9F 8F (translate 2F 0F (0F  []) 0)
               mkMerge 1F 2F (translate 0F 0F (0F  []) 0)
               mkMerge 7F 6F (translate 1F 0F (0F  []) 1)
               mkMerge 15F 13F (translate 2F 0F (0F  []) 2)
               mkMerge 14F 15F (translate 2F 0F (0F  []) 2)
               mkMerge 5F 3F (translate 0F 0F (0F  []) 1)
               mkMerge 12F 10F (translate 1F 0F (0F  []) 2)
               mkMerge 4F 5F (translate 0F 0F (0F  []) 2)
               mkMerge 11F 12F (translate 1F 0F (0F  []) 3)
               mkMerge 1F 9F (translate 2F 0F (0F  []) 4)
               mkMerge 0F 1F (translate 0F 0F (0F  []) 0)
               mkMerge 0F 7F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 15F 13F (seed 0)
               mkMerge 5F 3F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 10F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 8F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 2F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 6F (translate 1F 0F (0F  []) 2)
               mkMerge 2F 0F (translate 0F 0F (0F  []) 2)
               mkMerge 6F 0F (translate 1F 0F (0F  []) 3)
               mkMerge 13F 14F (translate 2F 0F (0F  []) 4)
               mkMerge 3F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 11F (translate 1F 0F (0F  []) 1)
               [] )
             ( mkMerge 15F 14F (seed 0)
               mkMerge 5F 4F (translate 0F 0F (0F  []) 0)
               mkMerge 12F 11F (translate 1F 0F (0F  []) 1)
               mkMerge 9F 1F (translate 2F 0F (0F  []) 2)
               mkMerge 8F 9F (translate 2F 0F (0F  []) 2)
               mkMerge 1F 0F (translate 0F 0F (0F  []) 1)
               mkMerge 7F 0F (translate 1F 0F (0F  []) 2)
               mkMerge 2F 1F (translate 0F 0F (0F  []) 2)
               mkMerge 6F 7F (translate 1F 0F (0F  []) 3)
               mkMerge 13F 15F (translate 2F 0F (0F  []) 4)
               mkMerge 3F 5F (translate 0F 0F (0F  []) 0)
               mkMerge 10F 12F (translate 1F 0F (0F  []) 1)
               [] )
             []
             [] )
           []

  joinTrᵛ : Vec (Vec (Trace 16 3 arOf) 7) 7
  joinTrᵛ = ( []
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             [] )
           ( ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               mkMerge 3F 0F (seed 12)
               mkMerge 10F 0F (seed 15)
               mkMerge 13F 1F (seed 18)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               mkMerge 3F 1F (seed 12)
               mkMerge 10F 7F (seed 15)
               mkMerge 13F 9F (seed 18)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               mkMerge 3F 2F (seed 12)
               mkMerge 10F 6F (seed 15)
               mkMerge 13F 8F (seed 18)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               mkMerge 13F 10F (seed 22)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 14F 13F (seed 10)
               mkMerge 15F 13F (seed 11)
               mkMerge 3F 0F (seed 14)
               mkMerge 10F 0F (seed 21)
               mkMerge 13F 0F (seed 24)
               [] )
             [] )
           ( ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 13)
               mkMerge 7F 0F (seed 14)
               mkMerge 8F 0F (seed 15)
               mkMerge 9F 0F (seed 16)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               mkMerge 3F 1F (seed 9)
               mkMerge 4F 0F (seed 10)
               mkMerge 10F 7F (seed 12)
               mkMerge 11F 0F (seed 13)
               mkMerge 13F 9F (seed 15)
               mkMerge 14F 1F (seed 16)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               mkMerge 3F 2F (seed 9)
               mkMerge 4F 1F (seed 10)
               mkMerge 10F 6F (seed 12)
               mkMerge 11F 7F (seed 13)
               mkMerge 13F 8F (seed 15)
               mkMerge 14F 9F (seed 16)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 13)
               mkMerge 7F 0F (seed 14)
               mkMerge 8F 0F (seed 15)
               mkMerge 9F 0F (seed 16)
               [] )
             ( mkMerge 3F 0F (seed 0)
               mkMerge 4F 2F (seed 1)
               mkMerge 5F 1F (seed 2)
               mkMerge 10F 0F (seed 3)
               mkMerge 11F 6F (seed 4)
               mkMerge 12F 7F (seed 5)
               mkMerge 13F 1F (seed 6)
               mkMerge 14F 8F (seed 7)
               mkMerge 15F 9F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 14)
               mkMerge 7F 0F (seed 15)
               mkMerge 8F 0F (seed 16)
               mkMerge 9F 0F (seed 17)
               [] )
             [] )
           ( ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 13)
               mkMerge 7F 0F (seed 14)
               mkMerge 8F 0F (seed 15)
               mkMerge 9F 0F (seed 16)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               mkMerge 3F 0F (seed 9)
               mkMerge 4F 2F (seed 10)
               mkMerge 10F 0F (seed 12)
               mkMerge 11F 6F (seed 13)
               mkMerge 13F 1F (seed 15)
               mkMerge 14F 8F (seed 16)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               mkMerge 3F 2F (seed 9)
               mkMerge 4F 1F (seed 10)
               mkMerge 10F 6F (seed 12)
               mkMerge 11F 7F (seed 13)
               mkMerge 13F 8F (seed 15)
               mkMerge 14F 9F (seed 16)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 13)
               mkMerge 7F 0F (seed 14)
               mkMerge 8F 0F (seed 15)
               mkMerge 9F 0F (seed 16)
               [] )
             ( mkMerge 3F 1F (seed 0)
               mkMerge 4F 0F (seed 1)
               mkMerge 5F 2F (seed 2)
               mkMerge 10F 7F (seed 3)
               mkMerge 11F 0F (seed 4)
               mkMerge 12F 6F (seed 5)
               mkMerge 13F 9F (seed 6)
               mkMerge 14F 1F (seed 7)
               mkMerge 15F 8F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 14)
               mkMerge 7F 0F (seed 15)
               mkMerge 8F 0F (seed 16)
               mkMerge 9F 0F (seed 17)
               [] )
             [] )
           ( ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 13)
               mkMerge 7F 0F (seed 14)
               mkMerge 8F 0F (seed 15)
               mkMerge 9F 0F (seed 16)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               mkMerge 3F 0F (seed 9)
               mkMerge 4F 2F (seed 10)
               mkMerge 10F 0F (seed 12)
               mkMerge 11F 6F (seed 13)
               mkMerge 13F 1F (seed 15)
               mkMerge 14F 8F (seed 16)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               mkMerge 3F 1F (seed 9)
               mkMerge 4F 0F (seed 10)
               mkMerge 10F 7F (seed 12)
               mkMerge 11F 0F (seed 13)
               mkMerge 13F 9F (seed 15)
               mkMerge 14F 1F (seed 16)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 13)
               mkMerge 7F 0F (seed 14)
               mkMerge 8F 0F (seed 15)
               mkMerge 9F 0F (seed 16)
               [] )
             ( mkMerge 3F 2F (seed 0)
               mkMerge 4F 1F (seed 1)
               mkMerge 5F 0F (seed 2)
               mkMerge 10F 6F (seed 3)
               mkMerge 11F 7F (seed 4)
               mkMerge 12F 0F (seed 5)
               mkMerge 13F 8F (seed 6)
               mkMerge 14F 9F (seed 7)
               mkMerge 15F 1F (seed 8)
               mkMerge 1F 0F (seed 9)
               mkMerge 2F 0F (seed 10)
               mkMerge 6F 0F (seed 14)
               mkMerge 7F 0F (seed 15)
               mkMerge 8F 0F (seed 16)
               mkMerge 9F 0F (seed 17)
               [] )
             [] )
           ( ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               mkMerge 3F 0F (seed 13)
               mkMerge 10F 0F (seed 16)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               mkMerge 3F 1F (seed 13)
               mkMerge 10F 7F (seed 16)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               mkMerge 3F 2F (seed 13)
               mkMerge 10F 6F (seed 16)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 4F 3F (seed 2)
               mkMerge 5F 3F (seed 3)
               mkMerge 6F 0F (seed 4)
               mkMerge 7F 0F (seed 5)
               mkMerge 8F 0F (seed 6)
               mkMerge 9F 0F (seed 7)
               mkMerge 11F 10F (seed 8)
               mkMerge 12F 10F (seed 9)
               mkMerge 13F 10F (seed 10)
               mkMerge 14F 10F (seed 11)
               mkMerge 15F 10F (seed 12)
               mkMerge 3F 0F (seed 15)
               mkMerge 10F 0F (seed 22)
               [] )
             [] )
           ( ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             ( mkMerge 1F 0F (seed 0)
               mkMerge 2F 0F (seed 1)
               mkMerge 3F 0F (seed 2)
               mkMerge 4F 0F (seed 3)
               mkMerge 5F 0F (seed 4)
               mkMerge 6F 0F (seed 5)
               mkMerge 7F 0F (seed 6)
               mkMerge 8F 0F (seed 7)
               mkMerge 9F 0F (seed 8)
               mkMerge 10F 0F (seed 9)
               mkMerge 11F 0F (seed 10)
               mkMerge 12F 0F (seed 11)
               mkMerge 13F 0F (seed 12)
               mkMerge 14F 0F (seed 13)
               mkMerge 15F 0F (seed 14)
               [] )
             [] )
           []

The verification

One decision for the whole certificate, one for the meet-table match; both compute to yes — that computation is the re-verification of every engine claim above.

open LatticeCheck 𝑭 𝑺

certOK : LatticeCertOK cert
certOK = from-yes (latticeCertOK? cert)

certMeet : MeetMatches 𝑭 𝑺 𝑳 cert
certMeet = from-yes (meetMatches? 𝑭 𝑺 𝑳 cert)

The headline theorems

The target lattice is decidably representable, witnessed by this algebra; and the certificate's congruence list is a complete Layer-D enumeration of the algebra's decidable congruences.

SLR09-Representableᵈ : Representableᵈ (toLattice 𝑳)
SLR09-Representableᵈ = certRepresentableᵈ 𝑭 𝑺 𝑳 cert certOK certMeet

SLR09-FiniteCongruencesᵈ : FiniteCongruencesᵈ 𝑨
SLR09-FiniteCongruencesᵈ = certFiniteCongruencesᵈ cert certOK