Section SystemUnification.
Variable (X: Const).
Definition eq: Type := exp X * exp X.
Definition eqs := list eq.
Implicit Types (sigma: fin -> exp X) (e: eq) (E : eqs).
Definition eq_typing  Gamma e A := Gamma ⊢ fst e : A /\ Gamma ⊢ snd e : A.
Notation "Gamma ⊢₂ e : A" := (eq_typing Gamma e A) (at level 80, e at level 99).
Reserved Notation  "Gamma ⊢₊₊ E : L" (at level 80, E at level 99).
Inductive eqs_typing Gamma : eqs -> list type -> Prop :=
| eqs_typing_nil: Gamma ⊢₊₊ nil : nil
| eqs_typing_cons s t A E L: Gamma ⊢ s : A -> Gamma ⊢ t : A -> Gamma ⊢₊₊ E : L -> Gamma ⊢₊₊ ((s,t) :: E) : A :: L
where  "Gamma ⊢₊₊ E : L" := (eqs_typing Gamma E L).
Hint Constructors eqs_typing : core.
Definition left_side E := map fst E.
Definition right_side E := map snd E.
Hint Resolve left_typing right_typing : core.
Definition vars' (e: eq) := vars (fst e) ++ vars (snd e).
Definition Vars' (E: list eq) := flat_map vars' E.
Hint Rewrite Vars'_cons Vars'_app : simplify.
Definition subst_eq sigma e := let (s, t) := e in (sigma • s, sigma • t).
Notation "sigma •₊₊ E" := (map (subst_eq sigma) E) (at level 69, right associativity).
Hint Rewrite left_subst_eqs right_subst_eqs : simplify.
Hint Rewrite all_terms_cons_iff : simplify.
Hint Rewrite all_terms_app : simplify.
Class sysuni :=
{
Gammaᵤ'  : ctx;
Eᵤ'  : eqs;
Lᵤ'  : list type;
Hᵤ' : Gammaᵤ' ⊢₊₊ Eᵤ' : Lᵤ';
}.
Definition SU (I: sysuni) := exists (Delta: ctx) (sigma: fin -> exp X),
(Delta ⊩ sigma : Gammaᵤ') /\ forall s t, (s, t) ∈ Eᵤ' -> sigma • s ≡ sigma • t.
Arguments SU: clear implicits.
Hint Resolve Hᵤ' : core.
Definition linearize_terms (S: list (exp X)) :=
lambda AppR (var 0) (renL shift S).

Lemma linearize_terms_equiv S T:
linearize_terms S ≡ linearize_terms T <-> S ≡₊ T.