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ErasedRecognizers.v
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ErasedRecognizers.v
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From Equations Require Import Equations.
Require Export SystemFR.RedTactics.
Opaque reducible_values.
Lemma reducible_values_is_pair:
forall ρ v,
[ ρ ⊨ is_pair v : T_bool ]v.
Proof.
destruct v; repeat step || simp_red.
Qed.
Lemma reducible_values_is_succ:
forall ρ v,
[ ρ ⊨ is_succ v : T_bool ]v.
Proof.
destruct v; repeat step || simp_red.
Qed.
Lemma reducible_values_is_lambda:
forall ρ v,
[ ρ ⊨ is_lambda v : T_bool ]v.
Proof.
destruct v; repeat step || simp_red.
Qed.
Lemma reducible_is_pair:
forall ρ t,
valid_interpretation ρ ->
[ ρ ⊨ t : T_top ] ->
[ ρ ⊨ boolean_recognizer 0 t : T_bool ].
Proof.
unfold reduces_to; repeat step.
exists (is_pair v); steps; eauto using reducible_values_is_pair.
eapply star_trans; eauto with cbvlemmas.
apply star_one.
constructor;
eauto using red_is_val;
eauto using fv_nil_top_level_var with fv.
Qed.
Lemma reducible_is_succ:
forall ρ t,
valid_interpretation ρ ->
[ ρ ⊨ t : T_top ] ->
[ ρ ⊨ boolean_recognizer 1 t : T_bool ].
Proof.
unfold reduces_to; repeat step.
exists (is_succ v); steps; eauto using reducible_values_is_succ.
eapply star_trans; eauto with cbvlemmas.
apply star_one.
constructor;
eauto using red_is_val;
eauto using fv_nil_top_level_var with fv.
Qed.
Lemma reducible_is_lambda:
forall ρ t,
valid_interpretation ρ ->
[ ρ ⊨ t : T_top ] ->
[ ρ ⊨ boolean_recognizer 2 t : T_bool ].
Proof.
unfold reduces_to; repeat step.
exists (is_lambda v); steps; eauto using reducible_values_is_lambda.
eapply star_trans; eauto with cbvlemmas.
apply star_one.
constructor;
eauto using red_is_val;
eauto using fv_nil_top_level_var with fv.
Qed.
Lemma open_reducible_is_pair:
forall Θ Γ t,
[ Θ; Γ ⊨ t : T_top ] ->
[ Θ; Γ ⊨ boolean_recognizer 0 t : T_bool ].
Proof.
unfold open_reducible; steps; eauto using reducible_is_pair.
Qed.
Lemma open_reducible_is_succ:
forall Θ Γ t,
[ Θ; Γ ⊨ t : T_top ] ->
[ Θ; Γ ⊨ boolean_recognizer 1 t : T_bool ].
Proof.
unfold open_reducible; steps; eauto using reducible_is_succ.
Qed.
Lemma open_reducible_is_lambda:
forall Θ Γ t,
[ Θ; Γ ⊨ t : T_top ] ->
[ Θ; Γ ⊨ boolean_recognizer 2 t : T_bool ].
Proof.
unfold open_reducible; steps; eauto using reducible_is_lambda.
Qed.