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From iris.heap_lang Require Export spawn.
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From iris.heap_lang Require Import proofmode notation.
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Import uPred.
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Definition parN : namespace := nroot .@ "par".

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Definition par : val :=
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  λ: "fs",
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    let: "handle" := spawn (Fst "fs") in
    let: "v2" := Snd "fs" #() in
    let: "v1" := join "handle" in
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    ("v1", "v2").
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Notation "e1 ||| e2" := (par (Pair (λ: <>, e1) (λ: <>, e2)))%E : expr_scope.
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Section proof.
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Context `{!heapG Σ, !spawnG Σ}.
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(* Notice that this allows us to strip a later *after* the two Ψ have been
   brought together.  That is strictly stronger than first stripping a later
   and then merging them, as demonstrated by [tests/joining_existentials.v].
   This is why these are not Texan triples. *)
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Lemma par_spec (Ψ1 Ψ2 : val  iProp Σ) e (f1 f2 : val) (Φ : val  iProp Σ) :
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  to_val e = Some (f1,f2)%V 
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  (WP f1 #() {{ Ψ1 }}  WP f2 #() {{ Ψ2 }} 
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     v1 v2, Ψ1 v1  Ψ2 v2 -  Φ (v1,v2)%V)
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   WP par e {{ Φ }}.
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Proof.
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  iIntros (?) "(Hf1 & Hf2 & HΦ)".
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  rewrite /par. wp_value. wp_let. wp_proj.
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  wp_apply (spawn_spec parN with "Hf1"); try wp_done; try solve_ndisj.
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  iIntros (l) "Hl". wp_let. wp_proj. wp_bind (f2 _).
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  iApply (wp_wand with "Hf2"); iIntros (v) "H2". wp_let.
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  wp_apply (join_spec with "[$Hl]"). iIntros (w) "H1".
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  iSpecialize ("HΦ" with "* [-]"); first by iSplitL "H1". by wp_let.
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Qed.
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Lemma wp_par (Ψ1 Ψ2 : val  iProp Σ)
    (e1 e2 : expr) `{!Closed [] e1, Closed [] e2} (Φ : val  iProp Σ) :
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  (WP e1 {{ Ψ1 }}  WP e2 {{ Ψ2 }} 
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    v1 v2, Ψ1 v1  Ψ2 v2 -  Φ (v1,v2)%V)
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   WP e1 ||| e2 {{ Φ }}.
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Proof.
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  iIntros "(H1 & H2 & H)". iApply (par_spec Ψ1 Ψ2 with "[- $H]"); try wp_done.
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  iSplitL "H1"; by wp_let.
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Qed.
End proof.