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ErasedPrimitive.v
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ErasedPrimitive.v
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Require Export SystemFR.ReducibilityOpenEquivalent.
Opaque reducible_values.
Opaque makeFresh.
#[export] Hint Rewrite PeanoNat.Nat.leb_le: primitives.
#[export] Hint Rewrite PeanoNat.Nat.leb_gt: primitives.
#[export] Hint Rewrite PeanoNat.Nat.ltb_lt: primitives.
#[export] Hint Rewrite PeanoNat.Nat.ltb_ge: primitives.
Opaque PeanoNat.Nat.leb.
Opaque PeanoNat.Nat.ltb.
Lemma last_step_unary_primitive:
forall v1 v o,
cbv_value v1 ->
cbv_value v ->
unary_primitive o v1 ~>* v ->
unary_primitive o v1 ~> v.
Proof.
intros.
inversion H1; clear H1.
destruct v; steps; step_inversion cbv_value.
force_invert scbv_step;
repeat star_smallstep_value || steps || constructor || no_step ; eauto with values smallstep ;
try solve [eapply scbv_step_same; [constructor |]; steps].
Qed.
Lemma last_step_binary_primitive:
forall v1 v2 v o,
cbv_value v1 ->
cbv_value v2 ->
cbv_value v ->
binary_primitive o v1 v2 ~>* v ->
binary_primitive o v1 v2 ~> v.
Proof.
intros.
inversion H2; clear H2.
destruct v; steps; step_inversion cbv_value.
force_invert scbv_step;
repeat star_smallstep_value || steps || constructor || no_step ; eauto with values smallstep ;
try solve [eapply scbv_step_same; [constructor |]; steps].
Qed.
Ltac last_step_binary_primitive :=
apply_anywhere last_step_binary_primitive; eauto with values;
force_invert scbv_step; repeat build_nat_inj || steps || autorewrite with primitives in *.
Ltac reducible_values_primitive :=
unfold reduces_to; intros ;
repeat simp_red ;
repeat is_nat_value_buildable ; steps ;
(unfold closed_term; repeat light || rewrite app_eq_nil_iff ; eauto 1 with fv wf erased) ||
(eexists; split; [ idtac | (apply star_one; constructor; try reflexivity )] ;
eauto using is_nat_value_build_nat).
Ltac reducible_primitive f :=
intros;
repeat top_level_unfold reduces_to || steps || simp_red || is_nat_value_buildable;
eapply star_backstep_reducible;
try apply star_smallstep_binary_primitive; eauto ;
repeat steps || list_utils || simp_red || apply f || eauto with values cbvlemmas ; t_closer.
(* Not *)
Lemma reducible_values_primitive_Not:
forall ρ v1,
[ ρ ⊨ v1 : T_bool ]v ->
[ ρ ⊨ unary_primitive Not v1 : T_bool ].
Proof.
unfold reduces_to;
repeat steps || simp_red; t_closer ;
solve [
eexists ; split ; [idtac | (apply star_one ; constructor ; try reflexivity)] ; steps ].
Qed.
Lemma reducible_primitive_Not:
forall ρ t1,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_bool ] ->
[ ρ ⊨ unary_primitive Not t1 : T_bool ].
Proof.
reducible_primitive reducible_values_primitive_Not.
Qed.
Lemma open_reducible_primitive_Not:
forall Θ Γ t1,
[ Θ;Γ ⊨ t1 : T_bool ] ->
[ Θ;Γ ⊨ unary_primitive Not t1 : T_bool ].
Proof. unfold open_reducible; steps; eauto using reducible_primitive_Not. Qed.
(* Plus *)
Lemma reducible_values_primitive_Plus:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Plus v1 v2 : T_nat ].
Proof. reducible_values_primitive. Qed.
Lemma reducible_primitive_Plus:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Plus t1 t2 : T_nat ].
Proof. reducible_primitive reducible_values_primitive_Plus. Qed.
Lemma open_reducible_primitive_Plus:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Plus t1 t2 : T_nat].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Plus. Qed.
(* Minus *)
Lemma reducible_values_primitive_Minus:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ binary_primitive Geq v1 v2 ≡ ttrue ] ->
[ ρ ⊨ binary_primitive Minus v1 v2 : T_nat ].
Proof.
reducible_values_primitive.
apply_anywhere equivalent_true.
last_step_binary_primitive.
Qed.
Lemma reducible_primitive_Minus:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ binary_primitive Geq t1 t2 ≡ ttrue ] ->
[ ρ ⊨ binary_primitive Minus t1 t2 : T_nat ].
Proof. reducible_primitive reducible_values_primitive_Minus.
eapply equivalent_trans; eauto.
apply equivalent_sym, equivalent_star; eauto with cbvlemmas smallstep wf fv; t_closer.
apply star_smallstep_binary_primitive; eauto with values.
Qed.
Lemma open_reducible_primitive_Minus:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Geq t1 t2 ≡ ttrue ] ->
[Θ; Γ ⊨ binary_primitive Minus t1 t2 : T_nat].
Proof. unfold open_reducible, open_equivalent; steps;
eauto using reducible_primitive_Minus.
Qed.
(* Mul *)
Lemma reducible_values_primitive_Mul:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Mul v1 v2 : T_nat ].
Proof. reducible_values_primitive. Qed.
Lemma reducible_primitive_Mul:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Mul t1 t2 : T_nat ].
Proof. reducible_primitive reducible_values_primitive_Mul. Qed.
Lemma open_reducible_primitive_Mul:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Mul t1 t2 : T_nat].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Mul. Qed.
(* Div *)
Lemma reducible_values_primitive_Div:
forall ρ v1 v2 ,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ binary_primitive Gt v2 zero ≡ ttrue] ->
[ ρ ⊨ binary_primitive Div v1 v2 : T_nat ].
Proof. reducible_values_primitive. apply_anywhere equivalent_true. last_step_binary_primitive. Qed.
Lemma reducible_primitive_Div:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ binary_primitive Gt t2 zero ≡ ttrue] ->
[ ρ ⊨ binary_primitive Div t1 t2 : T_nat ].
Proof.
reducible_primitive reducible_values_primitive_Div.
eapply equivalent_trans; eauto.
apply equivalent_sym, equivalent_star; eauto with cbvlemmas smallstep wf fv; t_closer.
Qed.
Lemma open_reducible_primitive_Div:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Gt t2 zero ≡ ttrue ] ->
[Θ; Γ ⊨ binary_primitive Div t1 t2 : T_nat].
Proof. unfold open_reducible, open_equivalent; steps ; eauto using reducible_primitive_Div. Qed.
(* Lt *)
Lemma reducible_values_primitive_Lt:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Lt v1 v2 : T_bool ].
Proof. reducible_values_primitive. steps. Qed.
Lemma reducible_primitive_Lt:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Lt t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Lt. Qed.
Lemma open_reducible_primitive_Lt:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Lt t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Lt. Qed.
(* Leq *)
Lemma reducible_values_primitive_Leq:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Leq v1 v2 : T_bool ].
Proof. reducible_values_primitive. steps. Qed.
Lemma reducible_primitive_Leq:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Leq t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Leq. Qed.
Lemma open_reducible_primitive_Leq:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Leq t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Leq. Qed.
(* Gt *)
Lemma reducible_values_primitive_Gt:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Gt v1 v2 : T_bool ].
Proof. reducible_values_primitive. steps. Qed.
Lemma reducible_primitive_Gt:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Gt t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Gt. Qed.
Lemma open_reducible_primitive_Gt:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Gt t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Gt. Qed.
(* Geq *)
Lemma reducible_values_primitive_Geq:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Geq v1 v2 : T_bool ].
Proof. reducible_values_primitive. steps. Qed.
Lemma reducible_primitive_Geq:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Geq t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Geq. Qed.
Lemma open_reducible_primitive_Geq:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Geq t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Geq. Qed.
(* Eq *)
Lemma reducible_values_primitive_Eq:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Eq v1 v2 : T_bool ].
Proof. reducible_values_primitive. steps. Qed.
Lemma reducible_primitive_Eq:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Eq t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Eq. Qed.
Lemma open_reducible_primitive_Eq:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Eq t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Eq. Qed.
(* Neq *)
Lemma reducible_values_primitive_Neq:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_nat ]v ->
[ ρ ⊨ v2 : T_nat ]v ->
[ ρ ⊨ binary_primitive Neq v1 v2 : T_bool ].
Proof. reducible_values_primitive. steps. Qed.
Lemma reducible_primitive_Neq:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_nat ] ->
[ ρ ⊨ t2 : T_nat ] ->
[ ρ ⊨ binary_primitive Neq t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Neq. Qed.
Lemma open_reducible_primitive_Neq:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_nat] ->
[Θ; Γ ⊨ t2 : T_nat] ->
[Θ; Γ ⊨ binary_primitive Neq t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Neq. Qed.
(* And *)
Lemma reducible_values_primitive_And:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_bool ]v ->
[ ρ ⊨ v2 : T_bool ]v ->
[ ρ ⊨ binary_primitive And v1 v2 : T_bool ].
Proof. reducible_values_primitive. Qed.
Lemma reducible_primitive_And:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_bool ] ->
[ ρ ⊨ t2 : T_bool ] ->
[ ρ ⊨ binary_primitive And t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_And. Qed.
Lemma open_reducible_primitive_And:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_bool] ->
[Θ; Γ ⊨ t2 : T_bool] ->
[Θ; Γ ⊨ binary_primitive And t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_And. Qed.
(* Or *)
Lemma reducible_values_primitive_Or:
forall ρ v1 v2,
[ ρ ⊨ v1 : T_bool ]v ->
[ ρ ⊨ v2 : T_bool ]v ->
[ ρ ⊨ binary_primitive Or v1 v2 : T_bool ].
Proof. reducible_values_primitive. Qed.
Lemma reducible_primitive_Or:
forall ρ t1 t2,
valid_interpretation ρ ->
[ ρ ⊨ t1 : T_bool ] ->
[ ρ ⊨ t2 : T_bool ] ->
[ ρ ⊨ binary_primitive Or t1 t2 : T_bool ].
Proof. reducible_primitive reducible_values_primitive_Or. Qed.
Lemma open_reducible_primitive_Or:
forall Θ Γ t1 t2,
[Θ; Γ ⊨ t1 : T_bool] ->
[Θ; Γ ⊨ t2 : T_bool] ->
[Θ; Γ ⊨ binary_primitive Or t1 t2 : T_bool].
Proof. unfold open_reducible; steps ; eauto using reducible_primitive_Or. Qed.
#[export]
Hint Resolve open_reducible_primitive_And: primitives.
#[export]
Hint Resolve open_reducible_primitive_Div: primitives.
#[export]
Hint Resolve open_reducible_primitive_Eq: primitives.
#[export]
Hint Resolve open_reducible_primitive_Geq: primitives.
#[export]
Hint Resolve open_reducible_primitive_Gt: primitives.
#[export]
Hint Resolve open_reducible_primitive_Leq: primitives.
#[export]
Hint Resolve open_reducible_primitive_Lt: primitives.
#[export]
Hint Resolve open_reducible_primitive_Minus: primitives.
#[export]
Hint Resolve open_reducible_primitive_Mul: primitives.
#[export]
Hint Resolve open_reducible_primitive_Neq: primitives.
#[export]
Hint Resolve open_reducible_primitive_Or: primitives.
#[export]
Hint Resolve open_reducible_primitive_Plus: primitives.
#[export]
Hint Resolve open_reducible_primitive_Not: primitives.