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81 lines
1.9 KiB
81 lines
1.9 KiB
From stdpp Require Import gmap base relations tactics.
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From iris Require Import prelude.
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From semantics.ts.systemf_mu Require Import lang notation types pure tactics.
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From Autosubst Require Import Autosubst.
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(** * Encoding of the untyped lambda calculus *)
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Definition D := (μ: #0 → #0)%ty.
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Definition lame (x : string) (e : expr) : val := RollV (λ: x, e).
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Definition appe (e1 e2 : expr) : expr := (unroll e1) e2.
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Lemma lame_typed n Γ x e :
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TY n; (<[x := D]> Γ) ⊢ e : D →
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TY n; Γ ⊢ lame x e : D.
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Proof. solve_typing. Qed.
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Lemma app_typed n Γ e1 e2 :
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TY n; Γ ⊢ e1 : D →
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TY n; Γ ⊢ e2 : D →
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TY n; Γ ⊢ appe e1 e2 : D.
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Proof.
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solve_typing.
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Qed.
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Lemma appe_step_l e1 e1' (v : val) :
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contextual_step e1 e1' →
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contextual_step (appe e1 v) (appe e1' v).
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Proof.
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intros Hstep. unfold appe.
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by eapply (fill_contextual_step [UnrollCtx; AppLCtx _]).
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Qed.
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Lemma appe_step_r e1 e2 e2' :
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contextual_step e2 e2' →
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contextual_step (appe e1 e2) (appe e1 e2').
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Proof.
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intros Hstep. unfold appe.
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by eapply (fill_contextual_step [AppRCtx _]).
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Qed.
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Lemma lame_step_beta x e (v : val) :
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rtc contextual_step (appe (lame x e) v) (lang.subst x v e).
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Proof.
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unfold appe, lame.
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econstructor 2.
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{ eapply (fill_contextual_step [AppLCtx _]).
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eapply base_contextual_step. by econstructor.
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}
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econstructor 2.
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{ eapply base_contextual_step. econstructor; eauto. }
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reflexivity.
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Qed.
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(* Divergence *)
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Definition ω : expr :=
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roll (λ: "x", (unroll "x") "x").
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Definition Ω := ((unroll ω) ω)%E.
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Lemma Ω_loops:
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tc contextual_step Ω Ω.
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Proof.
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econstructor 2; [|econstructor 1].
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- rewrite /Ω /ω. eauto.
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- rewrite /Ω. eauto.
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Qed.
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Lemma ω_typed :
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TY 0; ∅ ⊢ ω : D.
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Proof.
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solve_typing.
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Qed.
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Lemma Ω_typed : TY 0; ∅ ⊢ Ω : D.
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Proof.
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rewrite /Ω. econstructor.
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- eapply typed_unroll'; eauto using ω_typed.
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- eapply ω_typed.
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Qed.
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