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ErasedIte.v
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ErasedIte.v
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Require Export SystemFR.ErasedRefine.
Require Export SystemFR.ErasedTypeRefine.
Require Export SystemFR.ErasedSetOps.
Require Export SystemFR.TypeSugar.
Require Import Coq.Lists.List.
Opaque reducible_values.
Lemma open_reducible_T_ite:
forall Θ Γ T1 T2 b t1 t2 x,
wf t1 0 ->
wf t2 0 ->
wf T1 0 ->
wf T2 0 ->
is_erased_term t1 ->
is_erased_term t2 ->
is_erased_type T1 ->
is_erased_type T2 ->
subset (fv b) (support Γ) ->
subset (fv t1) (support Γ) ->
subset (fv t2) (support Γ) ->
subset (fv T1) (support Γ) ->
subset (fv T2) (support Γ) ->
~(x ∈ fv b) ->
~(x ∈ fv t1) ->
~(x ∈ fv t2) ->
~(x ∈ fv T1) ->
~(x ∈ fv T2) ->
~(x ∈ fv_context Γ) ->
~(x ∈ Θ) ->
is_erased_term b ->
is_erased_term t1 ->
is_erased_term t2 ->
[ Θ; Γ ⊨ b : T_bool ] ->
[ Θ; (x, T_equiv b ttrue) :: Γ ⊨ t1 : T1 ] ->
[ Θ; (x, T_equiv b tfalse) :: Γ ⊨ t2 : T2 ] ->
[ Θ; Γ ⊨ ite b t1 t2 : T_ite b T1 T2 ].
Proof.
unfold open_reducible; repeat step || t_instantiate_sat3.
repeat apply reducible_union || step || top_level_unfold reduces_to || simp_red.
- left.
apply reducible_refine; repeat step || (rewrite open_none by t_closer);
try solve [ equivalent_star ];
t_closer.
eapply star_backstep_reducible; repeat step || list_utils; eauto with cbvlemmas;
t_closer.
eapply backstep_reducible; eauto with smallstep; repeat step || list_utils;
eauto with fv wf erased.
unshelve epose proof (H23 ρ ((x, uu) :: lterms) _ _);
repeat step || list_utils || apply SatCons || simp_red || t_substitutions;
equivalent_star.
- right.
apply reducible_refine; repeat step || (rewrite open_none by t_closer) || list_utils;
t_closer;
try solve [
apply equivalent_star; repeat step || list_utils; t_closer;
eapply star_trans; eauto using star_smallstep_ite_cond; eauto using star_one with smallstep
].
eapply star_backstep_reducible; repeat step || list_utils; eauto with cbvlemmas;
t_closer.
eapply backstep_reducible; eauto with smallstep; repeat step || list_utils;
eauto with fv wf erased.
unshelve epose proof (H24 ρ ((x, uu) :: lterms) _ _);
repeat step || list_utils || apply SatCons || simp_red || t_substitutions;
equivalent_star.
Qed.