Section FIN_DISCR_DEC_SAT_FIN_DEC_EQ_NOSYMS_SAT.
Variable (Σ : fo_signature) (HΣ : finite_t (syms Σ) + discrete (syms  Σ)).
End FIN_DISCR_DEC_SAT_FIN_DEC_EQ_NOSYMS_SAT.
(* If the relation symbols in Σ have all their arities upper bounded by n and Σunif n is the signature with the same functions symbols and relations symbols as Σ except that the arity of relations is uniformly equal to n, then there is a reduction  SAT(Σ,𝔽,ℂ) ---> SAT(Σunif n,𝔽,ℂ) *)
(* With Σ=(sy,re) a signature with an empty type of term symbols and where all the finitely many relations in re have the same uniform arity n there is a reduction - from finite & decidable SAT over Σ - to finite & decodable SAT over Σ=(re,{R)} where re become constants symbols and R is a single relation of arity n+1 *)
(* With Σ=(sy,re) a signature with a discrete type sy of term symbols and among them, only constant symbols, there is a reduction - from finite & decidable SAT over Σ - to finite & decodable SAT over (ø,re) *)
(*      SAT(∅,{R²},𝔽,ℂ) ---> SAT(∅,{R^(2+n)},𝔽,ℂ)           *)
(* If Σ contains an n-ary relational symbol then there is a reduction  SAT(∅,{R^n},𝔽,ℂ) ---> SAT(Σ,𝔽,ℂ)  *)

Theorem FSAT_RELn_ANY Σ n r : ar_rels Σ r = n -> FSAT (Σrel n) ⪯ᵢ FSAT Σ.