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NormalizationPi.v
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NormalizationPi.v
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Require Export SystemFR.SubtypePi.
Require Export SystemFR.SubtypeMisc.
Opaque reducible_values.
Lemma npi: forall ρ S S' T T',
valid_interpretation ρ ->
is_erased_type T ->
is_erased_type T' ->
wf T 1 ->
wf T' 1 ->
pfv T term_var = nil ->
pfv T' term_var = nil ->
[ ρ ⊨ S = S' ] ->
(forall a, [ ρ ⊨ a : S' ]v -> [ ρ ⊨ open 0 T a = open 0 T' a ]) ->
[ ρ ⊨ T_arrow S T = T_arrow S' T' ].
Proof.
intros.
unfold equivalent_types;
repeat step || simp_red_goal || rewrite reducibility_rewrite;
t_closer.
- eapply equivalent_types_reducible; eauto.
eapply reducible_app; eauto using reducible_value_expr; steps.
eapply equivalent_types_reducible_back; eauto using reducible_value_expr.
- eapply equivalent_types_reducible_back; eauto.
+ eapply reducible_app; eauto using reducible_value_expr; steps.
eapply equivalent_types_reducible; eauto using reducible_value_expr.
+ apply_any; eauto using equivalent_types_reducible_values.
Qed.
Lemma open_npi_helper: forall Θ Γ S S' T T' x,
is_erased_type T ->
is_erased_type T' ->
wf T 1 ->
wf T' 1 ->
subset (fv T) (support Γ) ->
subset (fv T') (support Γ) ->
~ x ∈ pfv S' term_var ->
~ x ∈ pfv_context Γ term_var ->
[ Θ; Γ ⊨ S = S' ] ->
[ Θ; (x, S') :: Γ ⊨ open 0 T (fvar x term_var) = open 0 T' (fvar x term_var) ] ->
[ Θ; Γ ⊨ T_arrow S T = T_arrow S' T' ].
Proof.
unfold open_equivalent_types; repeat step || apply npi; t_closer.
unshelve epose proof (H8 ρ ((x, a) :: l) _ _ _);
repeat step || apply SatCons || t_substitutions; t_closer.
Qed.
Lemma open_npi: forall Γ S S' T T' x,
is_erased_type T ->
is_erased_type T' ->
wf T 1 ->
wf T' 1 ->
subset (fv T) (support Γ) ->
subset (fv T') (support Γ) ->
~ x ∈ pfv S term_var ->
~ x ∈ pfv S' term_var ->
~ x ∈ pfv_context Γ term_var ->
[ Γ ⊫ S = S' ] ->
[ (x, S') :: Γ ⊫ open 0 T (fvar x term_var) = open 0 T' (fvar x term_var) ] ->
[ Γ ⊫ T_arrow S T = T_arrow S' T' ].
Proof.
eauto using open_npi_helper.
Qed.