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From stdpp Require Import gmap base relations.
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From iris Require Import prelude.
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From semantics.lib Require Export debruijn.
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From semantics.ts.systemf Require Import lang notation parallel_subst types bigstep tactics logrel.
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From Equations Require Import Equations.
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(** * Free Theorems *)
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Implicit Types
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(Δ : nat)
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(Γ : typing_context)
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(v : val)
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(α : var)
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(e : expr)
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(A : type).
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Lemma not_every_type_inhabited : ¬ ∃ e, TY 0; ∅ ⊢ e : (∀: #0).
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Proof.
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intros (e & [Htycl Hty]%sem_soundness).
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specialize (Hty ∅ δ_any). simp type_interp in Hty.
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destruct Hty as (v & Hb & Hv).
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{ constructor. }
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simp type_interp in Hv. destruct Hv as (e' & -> & Hcl & Ha).
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(* [specialize_sem_type] defines a semantic type, spawning a subgoal for the closedness sidecondition *)
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specialize_sem_type Ha with (λ _, False) as τ; first done.
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simp type_interp in Ha. destruct Ha as (v' & He' & Hv').
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simp type_interp in Hv'. simpl in Hv'.
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done.
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Qed.
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Lemma all_identity :
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∀ e,
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TY 0; ∅ ⊢ e : (∀: #0 → #0) →
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∀ v, is_closed [] v → big_step (e <> (of_val v)) v.
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Proof.
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intros e [Htycl Hty]%sem_soundness v0 Hcl_v0.
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specialize (Hty ∅ δ_any). simp type_interp in Hty.
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destruct Hty as (v & Hb & Hv).
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{ constructor. }
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simp type_interp in Hv. destruct Hv as (e' & -> & Hcl & Hv).
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specialize_sem_type Hv with (λ v, v = v0) as τ.
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{ intros v ->; done. }
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simp type_interp in Hv.
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destruct Hv as (v & He & Hv).
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simp type_interp in Hv.
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destruct Hv as (x & e'' & -> & Hcl' & Hv).
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specialize (Hv v0 ltac:(done)).
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simp type_interp in Hv. destruct Hv as (v & Hb' & Hv).
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simp type_interp in Hv; simpl in Hv. subst v.
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rewrite subst_map_empty in Hb.
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eauto using big_step_of_val.
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Qed.
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