FLRP.Certificates.Group.TG9x4TwoByTwo¶
A machine-checked representation: Con(coset action of TransitiveGroup[9, 4] on 9 cosets of H (order 2)) ≅ 2x2 (four-element Boolean lattice)¶
This is the FLRP.Certificates.Group.TG9x4TwoByTwo module of the Agda Universal Algebra Library.
This module was emitted by scripts/python/flrp/emit_agda.py from
scripts/python/flrp/inputs/tg9x4_two_by_two.json. Do not edit it by hand; rerun the emitter instead.
The four-element Boolean lattice 2×2 as the upper interval [H, G] with G = TransitiveGroup(9,4) ≅ C3×S3 and H a point stabilizer (index 9), from the SmallLatticeReps manuscript Figure 10. Found by the issue #487 GAP subgroup-interval engine; this module is the direct-certificate route (coset action → claim → checker).
It re-verifies, end-to-end through the WP-6 certificate pipeline (#457), the
claim that the congruence lattice of the algebra "coset action of TransitiveGroup[9, 4] on 9 cosets of H (order 2)" — carrier size
9, unary operations g0, g1, g2 — is isomorphic to the lattice
"2x2 (four-element Boolean lattice)" (4 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.
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 9) 9) 3 opTables = (1F ∷ 3F ∷ 4F ∷ 0F ∷ 6F ∷ 7F ∷ 2F ∷ 8F ∷ 5F ∷ []) ∷ (2F ∷ 4F ∷ 5F ∷ 6F ∷ 7F ∷ 0F ∷ 8F ∷ 1F ∷ 3F ∷ []) ∷ (1F ∷ 0F ∷ 4F ∷ 3F ∷ 2F ∷ 7F ∷ 6F ∷ 5F ∷ 8F ∷ []) ∷ [] 𝑨 : Algebra {𝑆 = Sig-Unary (Fin 3)} 0ℓ 0ℓ 𝑨 = tablesToUnaryAlgebra 9 3 opTables 𝑭 : FiniteAlgebra 𝑨 𝑭 = tablesToUnaryAlgebra-FiniteAlgebra 9 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 "2x2 (four-element Boolean lattice)", 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 4 ∧-table = (0F ∷ 0F ∷ 0F ∷ 0F ∷ []) ∷ (0F ∷ 1F ∷ 0F ∷ 1F ∷ []) ∷ (0F ∷ 0F ∷ 2F ∷ 2F ∷ []) ∷ (0F ∷ 1F ∷ 2F ∷ 3F ∷ []) ∷ [] ∨-table = (0F ∷ 1F ∷ 2F ∷ 3F ∷ []) ∷ (1F ∷ 1F ∷ 3F ∷ 3F ∷ []) ∷ (2F ∷ 3F ∷ 2F ∷ 3F ∷ []) ∷ (3F ∷ 3F ∷ 3F ∷ 3F ∷ []) ∷ [] open FiniteLattice 𝑳 : FiniteLattice 𝑳 .size = 3 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 9 3 arOf 4 cert = mkLatticeCert partsᵛ 0F prinᵛ prinTrᵛ ∧-table ∨-table joinTrᵛ where partsᵛ : Vec (ParentVec 9) 4 partsᵛ = (0F ∷ 1F ∷ 2F ∷ 3F ∷ 4F ∷ 5F ∷ 6F ∷ 7F ∷ 8F ∷ []) ∷ (0F ∷ 0F ∷ 2F ∷ 0F ∷ 2F ∷ 5F ∷ 2F ∷ 5F ∷ 5F ∷ []) ∷ (0F ∷ 1F ∷ 0F ∷ 3F ∷ 1F ∷ 0F ∷ 3F ∷ 1F ∷ 3F ∷ []) ∷ (0F ∷ 0F ∷ 0F ∷ 0F ∷ 0F ∷ 0F ∷ 0F ∷ 0F ∷ 0F ∷ []) ∷ [] prinᵛ : Vec (Vec (Fin 4) 9) 9 prinᵛ = (0F ∷ 1F ∷ 2F ∷ 1F ∷ 3F ∷ 2F ∷ 3F ∷ 3F ∷ 3F ∷ []) ∷ (1F ∷ 0F ∷ 3F ∷ 1F ∷ 2F ∷ 3F ∷ 3F ∷ 2F ∷ 3F ∷ []) ∷ (2F ∷ 3F ∷ 0F ∷ 3F ∷ 1F ∷ 2F ∷ 1F ∷ 3F ∷ 3F ∷ []) ∷ (1F ∷ 1F ∷ 3F ∷ 0F ∷ 3F ∷ 3F ∷ 2F ∷ 3F ∷ 2F ∷ []) ∷ (3F ∷ 2F ∷ 1F ∷ 3F ∷ 0F ∷ 3F ∷ 1F ∷ 2F ∷ 3F ∷ []) ∷ (2F ∷ 3F ∷ 2F ∷ 3F ∷ 3F ∷ 0F ∷ 3F ∷ 1F ∷ 1F ∷ []) ∷ (3F ∷ 3F ∷ 1F ∷ 2F ∷ 1F ∷ 3F ∷ 0F ∷ 3F ∷ 2F ∷ []) ∷ (3F ∷ 2F ∷ 3F ∷ 3F ∷ 2F ∷ 1F ∷ 3F ∷ 0F ∷ 1F ∷ []) ∷ (3F ∷ 3F ∷ 3F ∷ 2F ∷ 3F ∷ 1F ∷ 2F ∷ 1F ∷ 0F ∷ []) ∷ [] prinTrᵛ : Vec (Vec (Trace 9 3 arOf) 9) 9 prinTrᵛ = ( [] ∷ ( mkMerge 0F 1F (seed 0) ∷ mkMerge 1F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 8F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 0F 2F (seed 0) ∷ mkMerge 1F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 6F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 4F 7F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 6F 8F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 0F 3F (seed 0) ∷ mkMerge 1F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 5F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 0F 4F (seed 0) ∷ mkMerge 1F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 2F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 3F 2F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 4F 8F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 0F 6F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 5F 1F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 0F 5F (seed 0) ∷ mkMerge 1F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 8F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 4F 1F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 6F 3F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 0F 6F (seed 0) ∷ mkMerge 1F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 6F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 3F 4F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 4F 5F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 0F 4F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 6F 7F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 0F 7F (seed 0) ∷ mkMerge 1F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 5F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 3F 5F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 4F 3F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 0F 8F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 6F 0F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 0F 8F (seed 0) ∷ mkMerge 1F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 2F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 8F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 3F 7F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 4F 0F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 0F 7F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 5F 6F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 1F 0F (seed 0) ∷ mkMerge 3F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 7F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ [] ∷ ( mkMerge 1F 2F (seed 0) ∷ mkMerge 3F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 4F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 0F 6F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 6F 7F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 3F 2F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 2F 8F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 1F 3F (seed 0) ∷ mkMerge 3F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 5F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 1F 4F (seed 0) ∷ mkMerge 3F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 2F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 6F 8F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 2F 5F (translate 2F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 1F 5F (seed 0) ∷ mkMerge 3F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 7F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 0F 8F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 6F 1F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 3F 5F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 7F 2F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 1F 6F (seed 0) ∷ mkMerge 3F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 6F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 0F 4F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 6F 5F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 3F 4F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 7F 3F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 1F 7F (seed 0) ∷ mkMerge 3F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 5F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 6F 3F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 2F 0F (translate 2F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 1F 8F (seed 0) ∷ mkMerge 3F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 4F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 8F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 0F 7F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 6F 0F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 3F 7F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 2F 3F (translate 2F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 2F 0F (seed 0) ∷ mkMerge 4F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 3F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 7F 4F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 8F 6F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 2F 1F (seed 0) ∷ mkMerge 4F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 0F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 6F 0F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 7F 6F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 2F 3F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 8F 2F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ [] ∷ ( mkMerge 2F 3F (seed 0) ∷ mkMerge 4F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 3F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 6F 1F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 7F 2F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 2F 1F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 0F 8F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 2F 4F (seed 0) ∷ mkMerge 4F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 3F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 2F 5F (seed 0) ∷ mkMerge 4F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 8F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 7F 1F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 8F 3F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 2F 6F (seed 0) ∷ mkMerge 4F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 0F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 2F 7F (seed 0) ∷ mkMerge 4F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 5F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 6F 5F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 7F 3F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 2F 8F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 0F 4F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 2F 8F (seed 0) ∷ mkMerge 4F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 5F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 8F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 6F 7F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 7F 0F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 2F 7F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 8F 1F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 3F 0F (seed 0) ∷ mkMerge 0F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 7F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 3F 1F (seed 0) ∷ mkMerge 0F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 8F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 3F 2F (seed 0) ∷ mkMerge 0F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 4F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 1F 6F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 2F 7F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 1F 2F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 8F 0F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ∷ ( mkMerge 3F 4F (seed 0) ∷ mkMerge 0F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 2F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 1F 2F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 2F 8F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 1F 6F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 6F 5F (translate 2F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 3F 5F (seed 0) ∷ mkMerge 0F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 7F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 1F 8F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 2F 1F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 1F 5F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 4F 3F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 3F 6F (seed 0) ∷ mkMerge 0F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 4F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 2F 5F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 4F 7F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 3F 7F (seed 0) ∷ mkMerge 0F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 5F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 1F 5F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 2F 3F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 1F 8F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 8F 4F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 3F 8F (seed 0) ∷ mkMerge 0F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 6F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 7F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 2F 0F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 4F 1F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 4F 0F (seed 0) ∷ mkMerge 6F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 1F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 2F 3F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 8F 4F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 6F 0F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 1F 5F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 4F 1F (seed 0) ∷ mkMerge 6F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 0F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 8F 6F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 5F 2F (translate 2F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 4F 2F (seed 0) ∷ mkMerge 6F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 1F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 4F 3F (seed 0) ∷ mkMerge 6F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 3F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 2F 1F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 8F 2F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 6F 1F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 5F 6F (translate 2F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ∷ ( mkMerge 4F 5F (seed 0) ∷ mkMerge 6F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 7F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 2F 8F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 8F 1F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 6F 5F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 5F 3F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 4F 6F (seed 0) ∷ mkMerge 6F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 0F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 4F 7F (seed 0) ∷ mkMerge 6F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 5F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 8F 3F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 5F 0F (translate 2F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 4F 8F (seed 0) ∷ mkMerge 6F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 7F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 8F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 2F 7F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 8F 0F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 6F 7F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 1F 6F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 5F 0F (seed 0) ∷ mkMerge 7F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 3F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 1F 4F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 3F 6F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 5F 1F (seed 0) ∷ mkMerge 7F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 0F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 8F 0F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 1F 6F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 5F 3F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 2F 7F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 5F 2F (seed 0) ∷ mkMerge 7F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 6F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 1F 7F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 3F 8F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 5F 3F (seed 0) ∷ mkMerge 7F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 3F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 8F 1F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 1F 2F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 5F 1F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 3F 4F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 5F 4F (seed 0) ∷ mkMerge 7F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 2F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 8F 2F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 1F 8F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 5F 6F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 3F 5F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ [] ∷ ( mkMerge 5F 6F (seed 0) ∷ mkMerge 7F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 6F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 8F 4F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 1F 5F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 5F 4F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 2F 3F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 5F 7F (seed 0) ∷ mkMerge 7F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 6F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 5F 8F (seed 0) ∷ mkMerge 7F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 0F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 1F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 2F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 6F 0F (seed 0) ∷ mkMerge 2F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 1F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 4F 3F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 5F 4F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 4F 0F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 7F 6F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 6F 1F (seed 0) ∷ mkMerge 2F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 0F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 4F 0F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 5F 6F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 4F 3F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 3F 7F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 6F 2F (seed 0) ∷ mkMerge 2F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 1F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 6F 3F (seed 0) ∷ mkMerge 2F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 1F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 5F 2F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 7F 4F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 6F 4F (seed 0) ∷ mkMerge 2F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 3F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 6F 5F (seed 0) ∷ mkMerge 2F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 7F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 4F 8F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 5F 1F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 4F 5F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 3F 2F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ∷ ( mkMerge 6F 7F (seed 0) ∷ mkMerge 2F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 5F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 4F 5F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 5F 3F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 4F 8F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 8F 0F (translate 2F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 6F 8F (seed 0) ∷ mkMerge 2F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 8F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 7F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 5F 0F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 7F 1F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 7F 0F (seed 0) ∷ mkMerge 8F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 1F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 5F 3F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 3F 4F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 8F 0F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 0F 6F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 7F 1F (seed 0) ∷ mkMerge 8F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 0F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 3F 6F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 0F 2F (translate 2F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 7F 2F (seed 0) ∷ mkMerge 8F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 4F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 5F 6F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 3F 7F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 8F 2F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 4F 0F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 7F 3F (seed 0) ∷ mkMerge 8F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 3F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 5F 1F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 3F 2F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 8F 1F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 4F 8F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 7F 4F (seed 0) ∷ mkMerge 8F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 2F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 3F 8F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 0F 5F (translate 2F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 7F 5F (seed 0) ∷ mkMerge 8F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 4F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 7F 6F (seed 0) ∷ mkMerge 8F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 5F 6F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 5F 4F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 3F 5F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 8F 4F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 0F 8F (translate 2F 0F (0F ∷ []) 4) ∷ [] ) ∷ [] ∷ ( mkMerge 7F 8F (seed 0) ∷ mkMerge 8F 5F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 1F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 3F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 4F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 2F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 8F 0F (seed 0) ∷ mkMerge 5F 1F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 1F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 7F 3F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 0F 4F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 7F 0F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 6F 5F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 8F 1F (seed 0) ∷ mkMerge 5F 3F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 0F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 7F 0F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 0F 6F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 7F 3F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 3F 2F (translate 2F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 8F 2F (seed 0) ∷ mkMerge 5F 4F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 5F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 4F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 7F 6F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 0F 7F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 7F 2F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 1F 8F (translate 1F 0F (0F ∷ []) 2) ∷ [] ) ∷ ( mkMerge 8F 3F (seed 0) ∷ mkMerge 5F 0F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 6F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 1F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 0F 2F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 1F 4F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 8F 4F (seed 0) ∷ mkMerge 5F 6F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 7F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 8F 2F (translate 2F 0F (0F ∷ []) 2) ∷ mkMerge 7F 2F (translate 0F 0F (0F ∷ []) 2) ∷ mkMerge 0F 8F (translate 1F 0F (0F ∷ []) 3) ∷ mkMerge 7F 6F (translate 2F 0F (0F ∷ []) 4) ∷ mkMerge 6F 1F (translate 1F 0F (0F ∷ []) 4) ∷ [] ) ∷ ( mkMerge 8F 5F (seed 0) ∷ mkMerge 5F 7F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 0F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 2F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 4F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 8F 6F (seed 0) ∷ mkMerge 5F 2F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 8F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 7F 4F (translate 0F 0F (0F ∷ []) 1) ∷ mkMerge 0F 5F (translate 1F 0F (0F ∷ []) 2) ∷ mkMerge 1F 7F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ ( mkMerge 8F 7F (seed 0) ∷ mkMerge 5F 8F (translate 0F 0F (0F ∷ []) 0) ∷ mkMerge 3F 1F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 0F 3F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 6F 4F (translate 1F 0F (0F ∷ []) 1) ∷ mkMerge 2F 6F (translate 1F 0F (0F ∷ []) 1) ∷ [] ) ∷ [] ∷ [] ) ∷ [] joinTrᵛ : Vec (Vec (Trace 9 3 arOf) 4) 4 joinTrᵛ = ( [] ∷ ( mkMerge 1F 0F (seed 0) ∷ mkMerge 3F 0F (seed 1) ∷ mkMerge 4F 2F (seed 2) ∷ mkMerge 6F 2F (seed 3) ∷ mkMerge 7F 5F (seed 4) ∷ mkMerge 8F 5F (seed 5) ∷ [] ) ∷ ( mkMerge 2F 0F (seed 0) ∷ mkMerge 4F 1F (seed 1) ∷ mkMerge 5F 0F (seed 2) ∷ mkMerge 6F 3F (seed 3) ∷ mkMerge 7F 1F (seed 4) ∷ mkMerge 8F 3F (seed 5) ∷ [] ) ∷ ( 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 1F 0F (seed 0) ∷ mkMerge 3F 0F (seed 1) ∷ mkMerge 4F 2F (seed 2) ∷ mkMerge 6F 2F (seed 3) ∷ mkMerge 7F 5F (seed 4) ∷ mkMerge 8F 5F (seed 5) ∷ [] ) ∷ ( mkMerge 1F 0F (seed 0) ∷ mkMerge 3F 0F (seed 1) ∷ mkMerge 4F 2F (seed 2) ∷ mkMerge 6F 2F (seed 3) ∷ mkMerge 7F 5F (seed 4) ∷ mkMerge 8F 5F (seed 5) ∷ [] ) ∷ ( mkMerge 1F 0F (seed 0) ∷ mkMerge 3F 0F (seed 1) ∷ mkMerge 4F 2F (seed 2) ∷ mkMerge 6F 2F (seed 3) ∷ mkMerge 7F 5F (seed 4) ∷ mkMerge 8F 5F (seed 5) ∷ mkMerge 2F 0F (seed 6) ∷ mkMerge 5F 0F (seed 8) ∷ [] ) ∷ ( mkMerge 1F 0F (seed 0) ∷ mkMerge 3F 0F (seed 1) ∷ mkMerge 4F 2F (seed 2) ∷ mkMerge 6F 2F (seed 3) ∷ mkMerge 7F 5F (seed 4) ∷ mkMerge 8F 5F (seed 5) ∷ mkMerge 2F 0F (seed 7) ∷ mkMerge 5F 0F (seed 10) ∷ [] ) ∷ [] ) ∷ ( ( mkMerge 2F 0F (seed 0) ∷ mkMerge 4F 1F (seed 1) ∷ mkMerge 5F 0F (seed 2) ∷ mkMerge 6F 3F (seed 3) ∷ mkMerge 7F 1F (seed 4) ∷ mkMerge 8F 3F (seed 5) ∷ [] ) ∷ ( mkMerge 2F 0F (seed 0) ∷ mkMerge 4F 1F (seed 1) ∷ mkMerge 5F 0F (seed 2) ∷ mkMerge 6F 3F (seed 3) ∷ mkMerge 7F 1F (seed 4) ∷ mkMerge 8F 3F (seed 5) ∷ mkMerge 1F 0F (seed 6) ∷ mkMerge 3F 0F (seed 7) ∷ [] ) ∷ ( mkMerge 2F 0F (seed 0) ∷ mkMerge 4F 1F (seed 1) ∷ mkMerge 5F 0F (seed 2) ∷ mkMerge 6F 3F (seed 3) ∷ mkMerge 7F 1F (seed 4) ∷ mkMerge 8F 3F (seed 5) ∷ [] ) ∷ ( mkMerge 2F 0F (seed 0) ∷ mkMerge 4F 1F (seed 1) ∷ mkMerge 5F 0F (seed 2) ∷ mkMerge 6F 3F (seed 3) ∷ mkMerge 7F 1F (seed 4) ∷ mkMerge 8F 3F (seed 5) ∷ mkMerge 1F 0F (seed 6) ∷ mkMerge 3F 0F (seed 8) ∷ [] ) ∷ [] ) ∷ ( ( 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 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 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 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) ∷ [] ) ∷ [] ) ∷ []
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.
TG9x4TwoByTwo-Representableᵈ : Representableᵈ (toLattice 𝑳) TG9x4TwoByTwo-Representableᵈ = certRepresentableᵈ 𝑭 𝑺 𝑳 cert certOK certMeet TG9x4TwoByTwo-FiniteCongruencesᵈ : FiniteCongruencesᵈ 𝑨 TG9x4TwoByTwo-FiniteCongruencesᵈ = certFiniteCongruencesᵈ cert certOK