From stdpp Require Import gmap base relations.
From iris Require Import prelude.
From semantics.ts.stlc_extended Require Import lang notation.

(** * Big-step semantics *)

Implicit Types
  (v : val)
  (e : expr).

Inductive big_step : expr → val → Prop :=
  | bs_lit (n : Z) :
    big_step (LitInt n) (LitIntV n)
  | bs_lam (x : binder) (e : expr) :
    big_step (λ: x, e)%E (λ: x, e)%V
  | bs_add e1 e2 (z1 z2 : Z) :
    big_step e1 (LitIntV z1) →
    big_step e2 (LitIntV z2) →
    big_step (Plus e1 e2) (LitIntV (z1 + z2))%Z
  | bs_app e1 e2 x e v2 v :
    big_step e1 (LamV x e) →
    big_step e2 v2 →
    big_step (subst' x (of_val v2) e) v →
    big_step (App e1 e2) v
  (* Pairs and fst, snd *)
  | bs_pair (e1 e2 : expr) (v1 v2 : val) :
    big_step e1 v1 →
    big_step e2 v2 →
    big_step (Pair e1 e2) (PairV v1 v2)
  | bs_fst (e : expr) (v1 v2: val):
    big_step e (PairV v1 v2) →
    big_step (Fst e) v1
  | bs_snd (e : expr) (v1 v2: val):
    big_step e (PairV v1 v2) →
    big_step (Fst e) v1
  (* Inj *)
  | bs_inj_l e v :
    big_step e v →
    big_step (InjL e) (InjLV v)
  | bs_inj_r e v :
    big_step e v →
    big_step (InjR e) (InjRV v)
  | bs_case_l (e_inj e_l e_r: expr) (v_inj v: val):
    big_step e_inj (InjLV v_inj) →
    big_step (e_l (of_val v_inj)) v →
    big_step (Case e_inj e_l e_r) v
  | bs_case_r (e_inj e_l e_r: expr) (v_inj v: val):
    big_step e_inj (InjRV v_inj) →
    big_step (e_r (of_val v_inj)) v →
    big_step (Case e_inj e_l e_r) v
.
#[export] Hint Constructors big_step : core.

Lemma big_step_of_val e v :
  e = of_val v →
  big_step e v.
Proof.
  intros ->.
  induction v; simpl; eauto.
Qed.


Lemma big_step_val v v' :
  big_step (of_val v) v' → v' = v.
Proof.
  enough (∀ e, big_step e v' → e = of_val v → v' = v) by naive_solver.
  intros e Hb.
  induction Hb in v |-*; intros Heq; subst; destruct v; inversion Heq; subst; naive_solver.
Qed.