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Firstorder.v
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Firstorder.v
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Require Import ssreflect ssrbool Eqdep_dec.
From Equations Require Import Equations.
From MetaCoq.Utils Require Import All_Forall MCPrelude MCSquash MCList utils.
From MetaCoq.Common Require Import Transform config Kernames.
From MetaCoq.PCUIC Require Import PCUICAst PCUICTyping PCUICFirstorder PCUICCasesHelper BDStrengthening PCUICCumulativity PCUICProgram.
From MetaCoq.Erasure Require Import EWcbvEval EWcbvEvalNamed.
From MetaCoq.ErasurePlugin Require Import Erasure ErasureCorrectness.
From MetaCoq.SafeChecker Require Import PCUICErrors PCUICWfEnv PCUICWfEnvImpl.
From Malfunction Require Import Malfunction Interpreter SemanticsSpec utils_array
Compile RealizabilitySemantics Pipeline PipelineCorrect.
Require Import ZArith Array.PArray List String Floats Lia Bool.
Import ListNotations.
From MetaCoq.Utils Require Import bytestring.
Import PCUICTransform (template_to_pcuic_transform, pcuic_expand_lets_transform).
Section firstorder.
Context {Σb : list (kername * bool)}.
Definition CoqType_to_camlType_oneind_type n k decl_type :
is_true (@firstorder_type Σb n k decl_type) -> camlType.
Proof.
unfold firstorder_type. destruct (fst _).
all: try solve [inversion 1].
(* case of tRel with a valid index *)
- intro; exact (Rel (n - S (n0-k))). (* n0 - k *)
(* case of tInd where Σb is saying that inductive_mind is first order *)
- intro; destruct ind. exact (Adt inductive_mind inductive_ind []).
Defined.
Fixpoint CoqType_to_camlType_ind_ctor n m cstr :
is_true
(alli
(fun (k : nat) '{| decl_type := t |} =>
@firstorder_type Σb m k t) n cstr) -> list camlType.
Proof.
destruct cstr; cbn; intros.
(* no more arguments *)
* exact [].
* apply andb_and in H. destruct H. destruct c.
(* we convert the type of the first type of the constructor *)
apply CoqType_to_camlType_oneind_type in H.
refine (H :: _).
(* we recursively convert *)
exact (CoqType_to_camlType_ind_ctor _ _ _ H0).
Defined.
Fixpoint CoqType_to_camlType_ind_ctors mind ind_ctors :
ind_params mind = [] ->
is_true (forallb (@firstorder_con Σb mind) ind_ctors) -> list (list camlType).
Proof.
destruct ind_ctors; intros Hparam H.
(* no more constructor *)
- exact [].
(* a :: ind_ctors constructors *)
- cbn in H. apply andb_and in H. destruct H.
refine (_ :: CoqType_to_camlType_ind_ctors _ _ Hparam H0). clear H0.
destruct c. unfold firstorder_con in H. cbn in H.
eapply CoqType_to_camlType_ind_ctor; eauto.
Defined.
Definition CoqType_to_camlType_oneind mind ind :
ind_params mind = [] ->
is_true (@firstorder_oneind Σb mind ind) -> list (list camlType).
Proof.
destruct ind. intros Hparam H. apply andb_and in H.
destruct H as [H _]. cbn in *. clear -H Hparam.
eapply CoqType_to_camlType_ind_ctors; eauto.
Defined.
Lemma bool_irr b (H H' : is_true b) : H = H'.
destruct b; [| inversion H].
assert (H = eq_refl).
eapply (K_dec (fun H => H = eq_refl)); eauto.
etransitivity; eauto.
eapply (K_dec (fun H => eq_refl = H)); eauto.
Unshelve. all: cbn; eauto.
all: intros b b'; pose (Coq.Bool.Bool.bool_dec b b'); intuition.
Qed.
Lemma CoqType_to_camlType_oneind_type_fo n k decl_type Hfo :
let T := CoqType_to_camlType_oneind_type n k decl_type Hfo in
is_true (negb (isArrow T)).
Proof.
unfold CoqType_to_camlType_oneind_type.
unfold firstorder_type in Hfo.
set (fst (PCUICAstUtils.decompose_app decl_type)) in *.
destruct t; inversion Hfo; cbn; eauto.
now destruct ind.
Qed.
Lemma CoqType_to_camlType_oneind_fo mind ind Hparam Hfo larg T :
In larg (CoqType_to_camlType_oneind mind ind Hparam Hfo) ->
In T larg -> is_true (negb (isArrow T)).
Proof.
destruct mind, ind. revert Hfo. induction ind_ctors0; intros Hfo Hlarg HT.
- cbn in *; destruct andb_and. destruct (a Hfo). destruct (in_nil Hlarg).
- cbn in Hlarg. destruct andb_and. destruct (a0 Hfo).
destruct andb_and. destruct (a1 i0). destruct a.
unfold In in Hlarg. destruct Hlarg.
2: { eapply IHind_ctors0; eauto. cbn. destruct andb_and.
destruct (a _). rewrite (bool_irr _ i6 i4). eauto.
Unshelve.
cbn. apply andb_and. split; cbn; eauto. }
cbn in Hparam, i3. rewrite Hparam in i3, H. rewrite app_nil_r in i3, H.
clear - i3 HT H. set (rev cstr_args0) in *. rewrite <- H in HT. revert T HT. clear H.
set (Datatypes.length ind_bodies0) in *. revert i3. generalize n 0. clear.
induction l; cbn; intros; [inversion HT|].
destruct andb_and. destruct (a0 i3). destruct a.
unfold In in HT. destruct HT.
+ rewrite <- H; eapply CoqType_to_camlType_oneind_type_fo; eauto.
+ eapply IHl. eauto.
Qed.
(* The following function morally do (map CoqType_to_camlType_oneind) while passing
the proof that everything is first order *)
Fixpoint CoqType_to_camlType_fix ind_bodies0
ind_finite0
ind_params0 ind_npars0 ind_universes0
ind_variance0 {struct ind_bodies0}:
ind_params0 = [] ->
forall ind_bodies1 : list one_inductive_body,
is_true
(forallb
(@firstorder_oneind Σb
{|
Erasure.P.PCUICEnvironment.ind_finite := ind_finite0;
Erasure.P.PCUICEnvironment.ind_npars := ind_npars0;
Erasure.P.PCUICEnvironment.ind_params := ind_params0;
Erasure.P.PCUICEnvironment.ind_bodies := ind_bodies1;
Erasure.P.PCUICEnvironment.ind_universes := ind_universes0;
Erasure.P.PCUICEnvironment.ind_variance := ind_variance0
|}) ind_bodies0) -> list (list (list camlType)).
Proof.
destruct ind_bodies0; cbn in *; intros.
- exact [].
- refine (CoqType_to_camlType_oneind _ _ _ _ :: CoqType_to_camlType_fix _ _ _ _ _ _ _ _ _); eauto.
2-3: unfold firstorder_oneind, firstorder_con; apply andb_and in H0; destruct H0; eauto.
eauto.
Defined.
Definition CoqType_to_camlType' mind :
ind_params mind = [] ->
is_true (forallb (@firstorder_oneind Σb mind) (ind_bodies mind)) ->
list (list (list camlType)).
Proof.
intros Hparam. destruct mind; unshelve eapply CoqType_to_camlType_fix; eauto.
Defined.
Lemma CoqType_to_camlType'_fo mind Hparam Hfo lind larg T :
In lind (CoqType_to_camlType' mind Hparam Hfo) ->
In larg lind ->
In T larg -> is_true (negb (isArrow T)).
Proof.
destruct mind. cbn in Hparam. unfold CoqType_to_camlType'. cbn in *.
revert Hfo.
enough (forall ind_bodies1
(Hfo : is_true
(forallb
(firstorder_oneind
{|
Erasure.P.PCUICEnvironment.ind_finite :=
ind_finite0;
Erasure.P.PCUICEnvironment.ind_npars := ind_npars0;
Erasure.P.PCUICEnvironment.ind_params :=
ind_params0;
Erasure.P.PCUICEnvironment.ind_bodies :=
ind_bodies1;
Erasure.P.PCUICEnvironment.ind_universes :=
ind_universes0;
Erasure.P.PCUICEnvironment.ind_variance :=
ind_variance0
|}) ind_bodies0)),
In lind
(CoqType_to_camlType_fix ind_bodies0 ind_finite0 ind_params0
ind_npars0 ind_universes0 ind_variance0 Hparam ind_bodies1 Hfo) ->
In larg lind -> In T larg -> is_true (negb (isArrow T))).
eapply H; eauto.
induction ind_bodies0; cbn in * ; [now intros |].
intros bodies1 Hfo Hlind Hlarg HT. unfold In in Hlind. destruct Hlind.
- rewrite <- H in Hlarg. eapply CoqType_to_camlType_oneind_fo; eauto.
- eapply IHind_bodies0; eauto.
Qed.
Lemma CoqType_to_camlType'_fo_alt T1 T2 Ts mind ind Hparam Hfo:
In Ts
(nth ind (CoqType_to_camlType'
mind Hparam Hfo) []) -> ~ In (T1 → T2) Ts.
Proof.
intros H H0.
epose proof (CoqType_to_camlType'_fo _ _ _ _ _ _ _ _ H0).
now cbn in H1. Unshelve. all:eauto.
case_eq (Nat.ltb ind (List.length (CoqType_to_camlType' mind
Hparam Hfo))); intro Hleq.
- apply nth_In. apply leb_complete in Hleq. lia.
- apply leb_complete_conv in Hleq.
assert (Datatypes.length (CoqType_to_camlType' mind Hparam Hfo) <= ind) by lia.
pose proof (nth_overflow _ [] H1).
rewrite H2 in H. destruct (in_nil H).
Qed.
Lemma CoqType_to_camlType'_length mind Hparam Hfo :
List.length (CoqType_to_camlType' mind Hparam Hfo)
= List.length (ind_bodies mind).
Proof.
destruct mind. cbn in *. revert Hfo.
enough (forall ind_bodies1
(Hfo : is_true
(forallb
(firstorder_oneind
{|
Erasure.P.PCUICEnvironment.ind_finite :=
ind_finite0;
Erasure.P.PCUICEnvironment.ind_npars :=
ind_npars0;
Erasure.P.PCUICEnvironment.ind_params :=
ind_params0;
Erasure.P.PCUICEnvironment.ind_bodies :=
ind_bodies1;
Erasure.P.PCUICEnvironment.ind_universes :=
ind_universes0;
Erasure.P.PCUICEnvironment.ind_variance :=
ind_variance0
|}) ind_bodies0)),
#|CoqType_to_camlType_fix
ind_bodies0 ind_finite0
ind_params0 ind_npars0
ind_universes0 ind_variance0
Hparam ind_bodies1 Hfo| =
#|ind_bodies0|); eauto.
induction ind_bodies0; cbn; eauto.
intros. f_equal. destruct andb_and.
destruct (a0 Hfo). eauto.
Qed.
Lemma CoqType_to_camlType_oneind_length mind Hparam x Hfo:
List.length
(CoqType_to_camlType_oneind
mind x Hparam Hfo) =
List.length (ind_ctors x).
Proof.
destruct x; cbn. destruct andb_and.
destruct (a Hfo). clear.
induction ind_ctors0; cbn; eauto.
destruct andb_and. destruct (a0 i0).
destruct a. cbn. f_equal. eapply IHind_ctors0.
Qed.
Lemma CoqType_to_camlType'_nth mind Hparam x Hfo Hfo' ind:
nth_error (ind_bodies mind) ind = Some x ->
ind < Datatypes.length (ind_bodies mind) ->
nth ind
(CoqType_to_camlType'
mind
Hparam
Hfo) [] =
CoqType_to_camlType_oneind mind x Hparam Hfo'.
Proof.
destruct mind. revert ind. cbn in *. revert Hfo Hfo'.
enough (
forall ind_bodies1 (Hfo : is_true
(forallb
(firstorder_oneind
(Build_mutual_inductive_body
ind_finite0 ind_npars0
ind_params0 ind_bodies1
ind_universes0 ind_variance0)) ind_bodies0))
(Hfo' : is_true
(firstorder_oneind
(Build_mutual_inductive_body
ind_finite0 ind_npars0
ind_params0 ind_bodies1
ind_universes0 ind_variance0) x)) (ind : nat),
nth_error ind_bodies0 ind = Some x ->
ind < Datatypes.length ind_bodies0 ->
nth ind
(CoqType_to_camlType_fix ind_bodies0 ind_finite0
ind_params0 ind_npars0 ind_universes0
ind_variance0 Hparam ind_bodies1 Hfo) [] =
CoqType_to_camlType_oneind
(Build_mutual_inductive_body
ind_finite0 ind_npars0
ind_params0 ind_bodies1
ind_universes0 ind_variance0) x Hparam Hfo'); eauto.
induction ind_bodies0; intros. cbn in *; lia.
destruct ind; cbn.
- destruct andb_and. destruct (a0 Hfo).
cbn in H. inversion H. subst. f_equal. clear. apply bool_irr.
- eapply IHind_bodies0; eauto.
cbn in *; lia.
Qed.
Lemma CoqType_to_camlType'_nth_length mind Hparam Hfo ind x:
nth_error (ind_bodies mind) ind = Some x ->
List.length
(nth ind
(CoqType_to_camlType'
mind
Hparam
Hfo) []) =
List.length (ind_ctors x).
Proof.
intros; erewrite CoqType_to_camlType'_nth; eauto.
eapply CoqType_to_camlType_oneind_length; eauto.
eapply nth_error_Some_length; eauto.
Unshelve. eapply forallb_nth_error in Hfo.
erewrite H in Hfo. exact Hfo.
Qed.
Definition CoqType_to_camlType mind :
ind_params mind = [] ->
is_true (forallb (@firstorder_oneind Σb mind) (ind_bodies mind)) -> ADT :=
fun Hparam H => (0, CoqType_to_camlType' mind Hparam H).
Lemma CoqType_to_camlType_ind_ctors_nth
mind ind_ctors Hparam a k i :
nth_error (CoqType_to_camlType_ind_ctors mind ind_ctors Hparam i) k
= Some a ->
exists a', nth_error ind_ctors k = Some a' /\
exists i',
a = CoqType_to_camlType_ind_ctor 0 #|ind_bodies mind| (rev (cstr_args a' ++ ind_params mind)) i'.
Proof.
revert a k i.
induction ind_ctors; cbn; intros.
- destruct k; inversion H.
- destruct andb_and. destruct (a1 _).
destruct k; destruct a.
+ inversion H. eexists; split; [reflexivity|].
cbn. unfold firstorder_con in i. cbn in i.
apply andb_and in i. destruct i as [i ?].
exists i. repeat f_equal. apply bool_irr.
+ eapply IHind_ctors; eauto.
Qed.
Lemma CoqType_to_camlType_ind_ctors_length
l n m i:
List.length (CoqType_to_camlType_ind_ctor n m l i) = List.length l.
revert n m i. induction l; cbn; eauto; intros.
destruct andb_and. destruct (a0 _). destruct a.
cbn. now f_equal.
Qed.
Lemma CoqType_to_camlType_ind_ctor_app
l l' n m i : { i' & { i'' & CoqType_to_camlType_ind_ctor n m (l++l') i = CoqType_to_camlType_ind_ctor n m l i' ++ CoqType_to_camlType_ind_ctor (#|l| + n) m l' i''}}.
pose proof (ii := i). rewrite alli_app in ii. eapply andb_and in ii. destruct ii as [i' i''].
exists i', i''. revert n i i' i''. induction l; cbn in *; intros.
- f_equal. eapply bool_irr.
- destruct andb_and. destruct (a0 _). destruct a.
destruct andb_and. destruct (a _). rewrite <- app_comm_cons; repeat f_equal.
* apply bool_irr.
* assert (S (#|l| + n) = #|l| + S n) by lia. revert i''. rewrite H. apply IHl.
Qed.
Lemma CoqType_to_camlType_oneind_nth_ctors mind Hparam x Hfo k l :
nth_error (CoqType_to_camlType_oneind mind x Hparam Hfo) k = Some l ->
exists i0, nth_error (CoqType_to_camlType_ind_ctors mind (ind_ctors x) Hparam i0) k = Some l.
Proof.
destruct x; cbn. destruct andb_and.
destruct (a Hfo). now eexists.
Qed.
Lemma CoqType_to_camlType_oneind_nth mind Hparam x Hfo k l :
nth_error (CoqType_to_camlType_oneind mind x Hparam Hfo) k = Some l
-> exists l', nth_error (ind_ctors x) k = Some l' /\ #|l| = #|cstr_args l'|.
Proof.
destruct x; cbn. destruct andb_and.
destruct (a Hfo). clear.
revert k l. induction ind_ctors0; cbn; eauto.
- intros k l H. rewrite nth_error_nil in H. inversion H.
- destruct andb_and. destruct (a0 i0).
destruct a. intros k l H. destruct k; cbn.
+ eexists; split; [reflexivity |]. cbn in *.
inversion H. rewrite CoqType_to_camlType_ind_ctors_length.
rewrite Hparam. rewrite app_nil_r. apply rev_length.
+ cbn in *. now eapply IHind_ctors0.
Qed.
Lemma CoqType_to_camlType_oneind_nth' mind Hparam x Hfo k l :
nth_error (ind_ctors x) k = Some l
-> exists l', nth_error (CoqType_to_camlType_oneind mind x Hparam Hfo) k = Some l' /\ #|l'| = #|cstr_args l|.
Proof.
destruct x; cbn. destruct andb_and.
destruct (a Hfo). clear.
revert k l. induction ind_ctors0; cbn; eauto.
- intros k l H. rewrite nth_error_nil in H. inversion H.
- destruct andb_and. destruct (a0 i0).
destruct a. intros k l H. destruct k; cbn.
+ eexists; split; [reflexivity |]. cbn in *.
inversion H. rewrite CoqType_to_camlType_ind_ctors_length.
rewrite Hparam. rewrite app_nil_r. apply rev_length.
+ cbn in *. now eapply IHind_ctors0.
Qed.
#[local] Instance cf_ : checker_flags := extraction_checker_flags.
#[local] Instance nf_ : PCUICSN.normalizing_flags := PCUICSN.extraction_normalizing.
Parameter Normalisation : forall Σ0 : PCUICAst.PCUICEnvironment.global_env_ext, PCUICTyping.wf_ext Σ0 -> PCUICSN.NormalizationIn Σ0.
Definition compile_pipeline `{Heap} (Σ:global_env_ext_map) t HΣ expΣ expt (typing : {T:term & ∥ Σ;;; [] |- t : T ∥}) :=
(@compile_malfunction_pipeline _ _ Σ t (projT1 typing) HΣ expΣ expt (projT2 typing) Normalisation).2.
Fixpoint wellformed_pure Σ Γ x :
CompileCorrect.wellformed Σ Γ x -> isPure x
with wellformed_binding_subset Σ Γ x :
CompileCorrect.wellformed_binding Σ Γ x -> isPure_binding x.
Proof.
2: destruct x. destruct x.
all: intros Hwf.
all: try solve [cbn in *; eauto].
- cbn in *. destruct p; rtoProp; eauto.
- cbn in *. destruct p; rtoProp; split; eauto. clear H.
induction l; eauto. cbn in *. rtoProp. split; eauto.
- cbn in *. destruct p; cbn in *; rtoProp; split; eauto. clear H H0. induction l; eauto.
cbn in *. rtoProp. split; eauto.
- cbn in *. destruct p; cbn in *; rtoProp; split; eauto. clear H. induction l; eauto.
cbn in *. rtoProp. split; eauto. destruct a. eauto.
- cbn in *. destruct p; rtoProp; eauto. destruct p; eauto.
- cbn in *. destruct p; rtoProp; eauto. repeat destruct p; eauto. rtoProp. split; eauto.
- cbn in *. destruct p; rtoProp; eauto. repeat destruct p; eauto.
- cbn in *. destruct p; rtoProp; eauto. clear H. induction l; eauto.
cbn in *. rtoProp. split; eauto.
- cbn in *. destruct p; rtoProp; eauto.
- cbn in *. destruct p; rtoProp; eauto.
- cbn in *. revert Hwf. set (_ ++ _). clearbody l0. revert l0. induction l; eauto; intros.
destruct a; cbn in *. rtoProp. split; eauto.
Qed.
Lemma compile_pure `{Heap} `{WcbvFlags} (Σ:global_env_ext_map) t HΣ expΣ expt typing :
is_true (isPure (compile_pipeline Σ t HΣ expΣ expt typing)).
Proof.
eapply wellformed_pure, verified_malfunction_pipeline_wellformed.
Qed.
Lemma compile_value_pure `{Heap} (Σ:global_env_ext) (wf: PCUICTyping.wf_ext Σ) Σ' t
(Hnparam : forall (i : kername) (mdecl : mutual_inductive_body),
lookup_env Σ i = Some (InductiveDecl mdecl) ->
ind_npars mdecl = 0):
firstorder_value Σ [] t ->
isPure_value (compile_value_mf' Σ Σ' t).
Proof.
revert t. eapply firstorder_value_inds. intros.
cbn. unfold compile_value_mf'. rewrite compile_value_box_mkApps. cbn.
unfold ErasureCorrectness.pcuic_lookup_inductive_pars, EGlobalEnv.lookup_constructor_pars_args.
rewrite PCUICExpandLetsCorrectness.trans_lookup.
unshelve eapply PCUICInductiveInversion.Construct_Ind_ind_eq' in X; eauto.
repeat destruct X as [? X]. repeat destruct d as [d ?]. unfold declared_minductive in d.
revert d; intro. cbn in d. unfold PCUICExpandLetsCorrectness.SE.lookup_env.
erewrite PCUICExpandLetsCorrectness.SE.lookup_global_Some_iff_In_NoDup in d.
2: { eapply NoDup_on_global_decls. destruct wf as [wf ?]. destruct wf. eauto. }
rewrite d. cbn. erewrite Hnparam. 2: now rewrite <- d.
rewrite skipn_0; destruct args; cbn.
- destruct Compile.lookup_constructor_args; cbn; eauto.
- destruct Compile.lookup_constructor_args; cbn; eauto.
inversion H2; subst. econstructor; eauto.
repeat rewrite map_map. now rewrite Forall_map.
Qed.
Definition isFunction_named (v : EWcbvEvalNamed.value) :=
match v with
EWcbvEvalNamed.vClos _ _ _ | EWcbvEvalNamed.vRecClos _ _ _ => true
| _ => false
end.
Lemma isFunction_isfunction_named t v : ErasureCorrectness.isFunction t ->
represents_value v (EImplementBox.implement_box t) ->
isFunction_named v.
Proof.
intros H H'; destruct t; inversion H; inversion H'; eauto.
Qed.
Lemma compile_function {P : Pointer} {HP : CompatiblePtr P P} {HHeap : @Heap P} `{@CompatibleHeap P P _ _ _} `{WcbvFlags}
(cf:=config.extraction_checker_flags)
(Σ:global_env_ext_map) h h' f v na A B HΣ expΣ
(Hax: Extract.axiom_free Σ)
(Hheap_refl : forall h, R_heap h h)
(wf : ∥ Σ ;;; [] |- f : tProd na A B ∥) expf :
∥Extract.nisErasable Σ [] f∥ ->
forall p,
let Σ_erase := (Transform.transform verified_named_erasure_pipeline (Σ, f) p).1 in
forall Σ' (HΣ' : CompileCorrect.malfunction_env_prop Σ_erase Σ')
(Hcons : forall (nm : Ident.t) (val val' : value),
In (nm, val) Σ' -> In (nm, val') Σ' -> vrel val val') ,
eval Σ' empty_locals h (compile_pipeline Σ f HΣ expΣ expf (_;wf)) h' v ->
isFunction v = true.
Proof.
intros Hnerase ? ? ? ? ? Heval. pose Normalisation.
unshelve epose proof (Hfunction := transform_erasure_pipeline_function' _ _ _ _ _).
6: eauto. all: eauto.
- destruct Hfunction as [v'[Heval' ?]].
unshelve eapply (Transform.preservation post_verified_named_erasure_pipeline) in Heval' as [v'' [Heval' [? [? ?]]]].
1: cbn; eauto.
Opaque post_verified_named_erasure_pipeline verified_erasure_pipeline.
destruct o as [? [? ?]]. destruct o as [? ?].
unfold implement_box_transformation, Transform.obseq in o1.
unfold name_annotation, Transform.obseq, Transform.run, time in o0.
sq. subst.
unfold compile_pipeline, compile_malfunction_pipeline, verified_malfunction_pipeline in Heval.
revert Heval; destruct_compose; intros.
unfold Transform.transform at 1 in Heval. cbn - [Transform.transform] in Heval.
unfold verified_named_erasure_pipeline in Heval.
revert Heval; destruct_compose; intros.
set (precond _ _ _ _ _ _ _ _) in Heval.
pose proof (ProofIrrelevance.proof_irrelevance _ p0 p). subst.
eapply CompileCorrect.compile_correct with (Σ' := Σ') (Γ' := empty_locals) (h:=h) in Heval'.
2: { intros. split; eapply assume_can_be_extracted; eauto. }
2: { intros. eauto. }
2: { intros. unfold verified_named_erasure_pipeline in Σ_erase.
revert Σ_erase HΣ'. destruct_compose; intros. eapply HΣ'. }
destruct H4.
eapply eval_det in Heval' as [? ?]; try eapply Heval; eauto.
2: { econstructor. }
clear Heval H2. eapply isFunction_isfunction_named in o0; eauto.
destruct v''; cbn in H5; try inversion o0; inversion H5; eauto.
Qed.
Opaque compile_pipeline compile_malfunction_pipeline verified_named_erasure_pipeline.
Definition irred Σ Γ t := forall t', PCUICReduction.red1 Σ Γ t t' -> False.
Lemma tConstruct_irred Σ Γ i n inst : irred Σ Γ (tConstruct i n inst).
Proof.
inversion 1. clear - H. destruct args; [inversion H|cbn in H].
revert H. generalize (tFix mfix idx) t. induction args; cbn; intros; eauto.
inversion H.
Qed.
Lemma compile_pipeline_tConstruct_nil : forall `{Heap}
kn ind n inst mind univ retro univ_decl
(Σ0 := mk_global_env univ [(kn , InductiveDecl mind)] retro),
let Σ : global_env_ext_map := (build_global_env_map Σ0, univ_decl) in
let i := mkInd kn ind in
forall HΣ expΣ expt
(fo : firstorder_ind Σ (firstorder_env Σ) i)
(Hnparam : forall (i : kername) (mdecl : mutual_inductive_body),
lookup_env Σ i = Some (InductiveDecl mdecl) ->
ind_npars mdecl = 0)
(Hlookup : lookup_env Σ kn = Some (InductiveDecl mind))
(wt : ∥ Σ;;; [] |- tConstruct i n inst : mkApps (tInd i inst) [] ∥),
let Σ_t := (Transform.transform verified_named_erasure_pipeline
(Σ, tConstruct i n inst)
(ErasureCorrectness.precond Σ (tConstruct i n inst)
(mkApps (tInd i inst) []) HΣ expΣ expt wt Normalisation)).1 in
forall Σ' (HΣ' : CompileCorrect.malfunction_env_prop Σ_t Σ')
(Hax : PCUICClassification.axiom_free Σ) h,
eval Σ' empty_locals h (compile_pipeline Σ (tConstruct i n inst) HΣ expΣ expt (existT _ _ wt))
h match Compile.lookup_constructor_args Σ_t i with
| Some num_args => let num_args_until_m := firstn n num_args in
let index := #| filter (fun x => match x with 0 => true | _ => false end) num_args_until_m| in
SemanticsSpec.value_Int (Malfunction.Int, BinInt.Z.of_nat index)
| None => fail "inductive not found"
end.
Proof.
intros.
unshelve epose proof (Hthm := verified_malfunction_pipeline_theorem H H0 Σ Hax HΣ expΣ
(tConstruct i n inst) expt (tConstruct i n inst) i inst [] wt fo Hnparam Normalisation _ _ _ Σ' HΣ' h);eauto.
{ eapply tConstruct_irred. }
cbn in Hthm.
unshelve epose proof (Hthm' := verified_named_erasure_pipeline_fo H H0 Σ HΣ expΣ
(tConstruct i n inst) expt (tConstruct i n inst) i inst [] fo Normalisation wt _ Hax);eauto.
eapply red_eval; eauto. eapply Normalisation.
{ eapply tConstruct_irred. }
sq. cbn in Hthm'.
unfold ErasureCorrectness.pcuic_lookup_inductive_pars, EGlobalEnv.lookup_constructor_pars_args in *.
rewrite PCUICExpandLetsCorrectness.trans_lookup in Hthm, Hthm'. cbn in Hthm, Hthm'.
rewrite ReflectEq.eqb_refl in Hthm, Hthm'. cbn in Hthm, Hthm'.
erewrite Hnparam, skipn_0 in Hthm; [|eauto].
inversion Hthm'; subst. clear Hthm'.
unfold EGlobalEnv.lookup_constructor_pars_args, Compile.lookup_constructor_args,
EGlobalEnv.lookup_constructor, EGlobalEnv.lookup_inductive,
EGlobalEnv.lookup_minductive in *. cbn in *.
set (EGlobalEnv.lookup_env _ _ ) in H3, Hthm.
case_eq o. 2: { intro X0. rewrite X0 in H3. inversion H3. }
intros ? eq. rewrite eq in Hthm, H3. pose proof (eq':=eq).
unfold o in eq'. eapply (verified_malfunction_pipeline_lookup H H0 Σ HΣ expΣ
(tConstruct i n inst) expt (tConstruct i n inst) i inst [] fo Normalisation wt _ Hax) in eq'.
rewrite eq'; eauto.
Qed.
Fixpoint map_forall {A B} {P : A -> Type} (f : forall a:A, P a -> B ) (l:list A) (Hl: All P l) : list B :=
match Hl with
All_nil => []
| All_cons x l HP Hl => f x HP :: map_forall f l Hl
end.
Definition isConstruct_ind t :=
match fst (PCUICAstUtils.decompose_app t) with
| tConstruct i _ _ => i
| _ => mkInd (MPfile [],"") 0
end.
Lemma typing_tConstruct_fo Σ args :
Forall (firstorder_value Σ []) args ->
Forall (fun t => exists inst pandi,
∥ Σ;;; [] |- t : mkApps (tInd (isConstruct_ind t) inst) pandi ∥) args.
Proof.
eapply Forall_impl. eapply firstorder_value_inds. intros.
unfold isConstruct_ind. rewrite PCUICAstUtils.decompose_app_mkApps; eauto.
Qed.
Lemma mkApps_irred Σ Γ t args :
(irred Σ Γ t) ->
(isLambda t -> False) ->
(forall mfix idx args, t = mkApps (tFix mfix idx) args -> False) ->
Forall (irred Σ Γ) args ->
irred Σ Γ (mkApps t args).
Proof.
revert t. induction args; eauto; cbn. intros t Ht Hlam Hfix Hargs X.
inversion Hargs; subst; clear Hargs. eapply IHargs; eauto.
{ clear X H2. intro X.
inversion 1; subst; eauto.
eapply (Hfix mfix idx (removelast args0)).
destruct args0. inversion H.
rewrite PCUICAstUtils.mkApps_nonempty in H; [inversion 1|].
inversion H; subst; eauto. }
intros. eapply (Hfix mfix idx (removelast args0)).
destruct args0. inversion H.
rewrite PCUICAstUtils.mkApps_nonempty in H; [inversion 1|].
inversion H; subst; eauto.
Qed.
Lemma tConstruct_tFix_discr i n inst (mfix : mfixpoint term) (idx : nat) (args : list term) :
tConstruct i n inst = mkApps (tFix mfix idx) args -> False.
Proof.
eapply rev_ind with (l := args); cbn; intros.
- inversion H.
- rewrite <- PCUICSafeReduce.tApp_mkApps in H0. inversion H0.
Qed.
Definition compile_pipeline_tConstruct_cons `{Heap} `{EWellformed.EEnvFlags}
kn ind n inst args mind univ retro univ_decl
(Σ0 := mk_global_env univ [(kn , InductiveDecl mind)] retro) :
let Σ : global_env_ext_map := (build_global_env_map Σ0, univ_decl) in
let i := mkInd kn ind in
forall HΣ expΣ expt
(fo : firstorder_ind Σ (firstorder_env Σ) i)
(Hnparam : forall (i : kername) (mdecl : mutual_inductive_body),
lookup_env Σ i = Some (InductiveDecl mdecl) ->
ind_npars mdecl = 0)
(Hlookup : lookup_env Σ kn = Some (InductiveDecl mind))
(Hargs: #|args| > 0)
(Hirred : Forall (irred Σ []) args)
(wt : ∥ Σ;;; [] |- mkApps (tConstruct i n inst) args : mkApps (tInd i inst) [] ∥),
let t := mkApps (tConstruct i n inst) args in
let Σ_t := (Transform.transform verified_named_erasure_pipeline
(Σ, t)
(ErasureCorrectness.precond Σ t
(mkApps (tInd i inst) []) HΣ expΣ expt wt Normalisation)).1 in
forall Σ' (HΣ' : CompileCorrect.malfunction_env_prop Σ_t Σ')
(Hax : PCUICClassification.axiom_free Σ) h,
let Σ_v := (Transform.transform
verified_named_erasure_pipeline
(Σ, t)
(ErasureCorrectness.precond2 Σ t
(mkApps (tInd i inst) []) HΣ expΣ expt wt
Normalisation t
(red_eval H H0 Σ Hax HΣ expΣ
(mkApps (tConstruct i n inst) args)
expt t i
inst [] wt fo Normalisation
(sq
(PCUICReduction.refl_red Σ []
(mkApps (tConstruct i n inst)
args)))
(mkApps_irred Σ []
(tConstruct i n inst) args
(tConstruct_irred Σ [] i n inst)
(fun
H2 : isLambda
(tConstruct i n inst) =>
ssrbool.not_false_is_true H2)
(tConstruct_tFix_discr i n inst)
Hirred)))).1 in
let vargs := map (compile_value_mf' Σ0 Σ_v) args in
eval Σ' empty_locals h (compile_pipeline Σ (mkApps (tConstruct i n inst) args) HΣ expΣ expt (existT _ _ wt))
h match Compile.lookup_constructor_args Σ_t i with
| Some num_args => let num_args_until_m := firstn n num_args in
let index := #| filter (fun x => match x with 0 => false | _ => true end) num_args_until_m| in
Block (int_of_nat index, vargs)
| None => fail "inductive not found"
end.
Proof.
intros.
unshelve epose proof (Hthm := verified_malfunction_pipeline_theorem H H0 Σ Hax HΣ expΣ
(mkApps (tConstruct i n inst) args) expt (mkApps (tConstruct i n inst) args) i inst [] wt fo Hnparam Normalisation _ _ _ _ HΣ' h);eauto.
{ eapply mkApps_irred; eauto. eapply tConstruct_irred. eapply tConstruct_tFix_discr. }
cbn in Hthm.
unshelve epose proof (Hthm' := verified_named_erasure_pipeline_fo H H0 Σ HΣ expΣ
(mkApps (tConstruct i n inst) args) expt (mkApps (tConstruct i n inst) args) i inst [] fo Normalisation wt _ Hax);eauto.
{ eapply red_eval; eauto. eapply Normalisation.
eapply mkApps_irred; eauto. eapply tConstruct_irred. eapply tConstruct_tFix_discr. }
sq.
unfold compile_value_mf' in Hthm, Hthm'. rewrite compile_value_box_mkApps in Hthm, Hthm'.
cbn in Hthm, Hthm'.
unfold ErasureCorrectness.pcuic_lookup_inductive_pars, EGlobalEnv.lookup_constructor_pars_args in *.
rewrite PCUICExpandLetsCorrectness.trans_lookup in Hthm, Hthm'. cbn in Hthm, Hthm'.
rewrite ReflectEq.eqb_refl in Hthm, Hthm'. cbn in Hthm, Hthm'.
erewrite Hnparam, skipn_0 in Hthm; [|eauto].
inversion Hthm'; subst. clear Hthm'.
unfold Compile.lookup_constructor_args, EGlobalEnv.lookup_constructor_pars_args,
EGlobalEnv.lookup_constructor, EGlobalEnv.lookup_inductive,
EGlobalEnv.lookup_minductive in *. cbn in *.
set (EGlobalEnv.lookup_env _ _ ) in H4, Hthm.
case_eq o. 2: { intro X0. rewrite X0 in H4. inversion H4. }
intros ? eq. rewrite eq in Hthm, H4. pose proof (eq':=eq).
unfold o in eq'. eapply (verified_malfunction_pipeline_lookup H H0 Σ HΣ expΣ
(mkApps (tConstruct i n inst) args) expt (mkApps (tConstruct i n inst) args) i inst [] fo Normalisation wt _ Hax) in eq'.
rewrite eq'; eauto.
destruct args; [inversion Hargs |].
destruct g; [inversion H4 |]. cbn in *.
destruct (nth_error _ _); [|inversion H4].
set (map _ (map _ _)) in Hthm. simpl in Hthm.
erewrite (f_equal (eval _ _ _ _ _)); eauto. clear Hthm. repeat f_equal.
unfold l. now erewrite map_map.
Qed.
Lemma filter_length_nil ind k mind Eind Hparam Hfo :
nth_error (ind_bodies mind) ind = Some Eind ->
List.length
(filter (fun x0 : nat => match x0 with
| 0 => true
| S _ => false
end)
(firstn k (map cstr_nargs (ind_ctors Eind)))) =
List.length
(filter (fun x : list camlType =>
match x with
| [] => true
| _ :: _ => false
end)
(firstn k
(nth ind
(CoqType_to_camlType' mind
Hparam Hfo) []))).
Proof.
intro Hind. unshelve erewrite CoqType_to_camlType'_nth; eauto.
- eapply nth_error_forallb in Hfo; eauto.
- destruct Eind; cbn. destruct andb_and. destruct (a _).
rewrite (filter_ext _ (fun x =>
match #|x| with
| 0 => true
| S _ => false
end)).
{ clear. intro l; induction l; eauto. }
revert k i0. clear Hind i i1 a. induction ind_ctors0; cbn; intros.
+ intros. now repeat rewrite firstn_nil.
+ destruct k; eauto; cbn. destruct andb_and. destruct (a0 _). cbn.
destruct a. cbn. rewrite CoqType_to_camlType_ind_ctors_length.
revert i1. cbn. rewrite Hparam app_nil_r. cbn.
rewrite rev_length. intro.
destruct cstr_args0; cbn.
* intros; f_equal. eapply IHind_ctors0; eauto.
* eapply IHind_ctors0.
- now eapply nth_error_Some_length.
Qed.
Lemma filter_length_not_nil ind k mind Eind Hparam Hfo :
nth_error (ind_bodies mind) ind = Some Eind ->
List.length
(filter (fun x0 : nat => match x0 with
| 0 => false
| S _ => true
end)
(firstn k (map cstr_nargs (ind_ctors Eind)))) =
List.length
(filter (fun x : list camlType =>
match x with
| [] => false
| _ :: _ => true
end)
(firstn k
(nth ind
(CoqType_to_camlType' mind
Hparam Hfo) []))).
Proof.
intro Hind. unshelve erewrite CoqType_to_camlType'_nth; eauto.
- eapply nth_error_forallb in Hfo; eauto.
- destruct Eind; cbn. destruct andb_and. destruct (a _).
rewrite (filter_ext _ (fun x =>
match #|x| with
| 0 => false
| S _ => true
end)).
{ clear. intro l; induction l; eauto. }
revert k i0. clear Hind i i1 a. induction ind_ctors0; cbn; intros.
+ intros. now repeat rewrite firstn_nil.
+ destruct k; eauto; cbn. destruct andb_and. destruct (a0 _). cbn.
destruct a. cbn. rewrite CoqType_to_camlType_ind_ctors_length.
revert i1. cbn. rewrite Hparam app_nil_r. cbn.
rewrite rev_length. intro.
destruct cstr_args0; cbn.
* eapply IHind_ctors0.
* intros; f_equal. eapply IHind_ctors0; eauto.
- now eapply nth_error_Some_length.
Qed.
End firstorder.
Opaque compile_pipeline compile_malfunction_pipeline verified_named_erasure_pipeline.
Lemma CoqType_to_camlType_oneind_type_Rel n k decl_type Hfo :
let T := CoqType_to_camlType_oneind_type (Σb := []) n k decl_type Hfo in
exists i, (k <= i) /\ (i < n + k) /\ T = Rel (n - S (i-k)).
Proof.
unfold CoqType_to_camlType_oneind_type.
unfold firstorder_type in Hfo.
set (fst (PCUICAstUtils.decompose_app decl_type)) in *.
set (snd (PCUICAstUtils.decompose_app decl_type)) in *.
destruct t; destruct l; inversion Hfo; cbn; eauto.
- apply andb_and in H0. destruct H0. apply leb_complete in H, H0.
exists n0; repeat split; eauto.
- destruct ind. inversion Hfo.
- destruct ind; inversion Hfo.
Qed.
Lemma CoqType_to_camlType_oneind_Rel mind ind larg T k Hfo Hparam :
In larg (CoqType_to_camlType_oneind (Σb := []) mind ind Hparam Hfo) ->
nth_error larg k = Some T ->
exists i, (k <= i) /\ (i < #|ind_bodies mind| + k) /\ T = Rel (#|ind_bodies mind| - S (i-k)).
Proof.
destruct mind, ind. revert k Hfo. induction ind_ctors0; intros k Hfo Hlarg HT.
- cbn in *; destruct andb_and. destruct (a Hfo). destruct (in_nil Hlarg).
- cbn; cbn in Hlarg. destruct andb_and. cbn in Hfo. destruct (a0 Hfo).
destruct andb_and. destruct (a1 i0). destruct a.
unfold In in Hlarg. destruct Hlarg.
2: { eapply IHind_ctors0; eauto. cbn. destruct andb_and.
destruct (a _). rewrite (bool_irr _ i6 i4). eauto.
Unshelve.
cbn. apply andb_and. split; cbn; eauto. }
cbn in Hparam, i3. rewrite Hparam in i3, H. rewrite app_nil_r in i3, H.
clear - i3 HT H. set (rev cstr_args0) in *. rewrite <- H in HT.
pose proof (nth_error_Some_length HT).
revert T HT. clear H H0. cbn; set (#|ind_bodies0|) in *.
setoid_rewrite <- (Nat.sub_0_r k) at 1. assert (k >= 0) by lia. revert H.
revert k i3. generalize 0 at 1 2 3 4.
induction l; cbn; intros. 1: { rewrite nth_error_nil in HT. inversion HT. }
destruct andb_and. destruct (a0 i3). destruct a.
cbn in HT.
case_eq (k - n0).
+ intro Heq.
rewrite Heq in HT. inversion HT.
unshelve epose (CoqType_to_camlType_oneind_type_Rel _ _ _ _).
5: erewrite H1 in e.
rewrite H1. destruct e as [i2 [? [? ?]]]. exists (i2 + k - n0).
repeat split. all: try lia. rewrite H3; f_equal. lia.
+ intros ? Heq. rewrite Heq in HT. cbn in HT. eapply IHl.
2: { assert (n1 = k - S n0) by lia. now rewrite H0 in HT. }
lia.
Qed.
Lemma CoqType_to_camlType'_unfold mind Σb Hparam Hfo l :
In l (CoqType_to_camlType' (Σb := Σb) mind Hparam Hfo) ->
exists ind Hfo', l = CoqType_to_camlType_oneind (Σb := Σb) mind ind Hparam Hfo'.
Proof.
destruct mind; cbn in *. revert Hfo.
enough (forall ind_bodies1, forall
Hfo : is_true
(forallb
(firstorder_oneind
{|
Erasure.P.PCUICEnvironment.ind_finite := ind_finite0;
Erasure.P.PCUICEnvironment.ind_npars := ind_npars0;
Erasure.P.PCUICEnvironment.ind_params := ind_params0;
Erasure.P.PCUICEnvironment.ind_bodies := ind_bodies1;
Erasure.P.PCUICEnvironment.ind_universes := ind_universes0;
Erasure.P.PCUICEnvironment.ind_variance := ind_variance0
|}) ind_bodies0),
In l
(CoqType_to_camlType_fix (Σb := Σb) ind_bodies0 ind_finite0 ind_params0 ind_npars0 ind_universes0 ind_variance0
Hparam ind_bodies1 Hfo) ->
exists
(ind : one_inductive_body) (Hfo' : is_true
(firstorder_oneind
{|
Erasure.P.PCUICEnvironment.ind_finite := ind_finite0;
Erasure.P.PCUICEnvironment.ind_npars := ind_npars0;
Erasure.P.PCUICEnvironment.ind_params := ind_params0;
Erasure.P.PCUICEnvironment.ind_bodies := ind_bodies1;
Erasure.P.PCUICEnvironment.ind_universes := ind_universes0;
Erasure.P.PCUICEnvironment.ind_variance := ind_variance0
|} ind)),
l =
CoqType_to_camlType_oneind (Σb := Σb)
{|
Erasure.P.PCUICEnvironment.ind_finite := ind_finite0;
Erasure.P.PCUICEnvironment.ind_npars := ind_npars0;
Erasure.P.PCUICEnvironment.ind_params := ind_params0;
Erasure.P.PCUICEnvironment.ind_bodies := ind_bodies1;
Erasure.P.PCUICEnvironment.ind_universes := ind_universes0;
Erasure.P.PCUICEnvironment.ind_variance := ind_variance0
|} ind Hparam Hfo').
{ eapply H; eauto. }
induction ind_bodies0; cbn; intros.
- inversion H.
- destruct andb_and. destruct (a0 Hfo). destruct H.
+ exists a. now exists i0.
+ eapply IHind_bodies0; eauto.
Qed.
Lemma CoqType_to_camlType'_Rel mind ind larg T k Hfo Hparam:
let typ := (CoqType_to_camlType' (Σb := []) mind Hparam Hfo) in
In larg (nth ind typ []) ->
nth_error larg k = Some T ->
exists i, (k <= i) /\ (i < #|ind_bodies mind| + k) /\ T = Rel (#|ind_bodies mind| - S (i-k)).
Proof.
intros typ. destruct (nth_in_or_default ind typ []).
2: { rewrite e. intros Hlarg; destruct (in_nil Hlarg). }
intro Hlarg. unfold typ in *. eapply CoqType_to_camlType'_unfold in i.
destruct i as [ind' [Hfo' i]]. rewrite i in Hlarg.
now unshelve eapply CoqType_to_camlType_oneind_Rel; eauto.
Qed.
(* Lemma CoqType_to_camlType'_Rel' mind ind Hparam Hfo larg T k :
let typ := (CoqType_to_camlType' (Σb := []) mind Hparam Hfo) in
In larg (nth ind typ []) ->
nth_error larg k = Some T -> exists i, (0 <= i) /\ (i < #| ind_bodies mind|) /\ T = Rel (#|ind_bodies mind| - S i).
Proof.
intros. unshelve epose (CoqType_to_camlType'_Rel _ ind); eauto.
edestruct e as [i [? [? ?]]]; eauto. exists (i - k). repeat split; eauto.
all: lia.
Qed. *)
Inductive nonDepProd : term -> Type :=
| nonDepProd_tInd : forall i u, nonDepProd (tInd i u)
| nonDepProd_Prod : forall na A B, closed A -> nonDepProd B -> nonDepProd (tProd na A B).
Lemma nonDep_closed t : nonDepProd t -> closed t.
Proof.
induction 1; cbn; eauto.
rewrite i; cbn. eapply PCUICLiftSubst.closed_upwards; eauto.
Qed.
Fixpoint nonDep_decompose (t:term) (HnonDep : nonDepProd t) : list term * term
:= match HnonDep with
| nonDepProd_tInd i u => ([], tInd i u)
| nonDepProd_Prod na A B clA ndB => let IH := nonDep_decompose B ndB in (A :: fst IH, snd IH)
end.
Lemma nonDep_typing_spine (cf:=config.extraction_checker_flags) (Σ:global_env_ext) us u_ty s (nonDep : nonDepProd u_ty) :
PCUICTyping.wf Σ ->
Σ ;;; [] |- u_ty : tSort s ->
All2 (fun u uty => Σ ;;; [] |- u : uty) us (fst (nonDep_decompose u_ty nonDep)) ->
PCUICGeneration.typing_spine Σ [] u_ty us (snd (nonDep_decompose u_ty nonDep)).
Proof.
revert s us. induction nonDep; cbn in *; intros s us wfΣ Hty; cbn.
- intro H; depelim H. econstructor; [|reflexivity]. repeat eexists; eauto.
- set (PCUICInversion.inversion_Prod Σ wfΣ Hty) in *.
destruct s0 as [s1 [s2 [HA [HB ?]]]]; cbn. intro H; depelim H.
epose (strengthening_type [] ([],, vass na A) [] B s2).
pose proof (nonDep_closed _ nonDep).
edestruct s0 as [s' [Hs _]]. cbn. rewrite PCUICLiftSubst.lift_closed; eauto.
econstructor; try reflexivity; eauto.
+ repeat eexists; eauto.
+ rewrite /subst1 PCUICClosed.subst_closedn; eauto.
Qed.
Lemma firstorder_con_notApp `{checker_flags} {Σb} mind a :
@firstorder_con Σb mind a ->
All (fun decl => ~ (isApp (decl_type decl))) (cstr_args a).
Proof.
intros Hfo. destruct a; cbn in *. induction cstr_args0; eauto.
cbn in Hfo. rewrite rev_app_distr in Hfo. cbn in Hfo.
rewrite alli_app in Hfo. rewrite <- plus_n_O in Hfo. apply andb_and in Hfo. destruct Hfo as [? Hfo].
econstructor.
- clear H0 IHcstr_args0. destruct a.
simpl in *. apply andb_and in Hfo. destruct Hfo as [Hfo _].
unfold firstorder_type in Hfo. cbn in Hfo.
assert (snd (PCUICAstUtils.decompose_app decl_type) = []).
{ destruct snd; eauto. destruct fst; try destruct ind; inversion Hfo. }
destruct decl_type; try solve [inversion Hfo]; eauto. intros _.
cbn in H0. pose proof (PCUICInduction.decompose_app_rec_length decl_type1 [decl_type2]).
rewrite H0 in H1. cbn in H1. lia.
- apply IHcstr_args0. now rewrite rev_app_distr.
Qed.
Lemma firstorder_type_closed kn l k T :
~ (isApp T) ->
firstorder_type (Σb := []) #|ind_bodies l| k T ->
closed (subst (inds kn [] (ind_bodies l)) k T@[[]]).
Proof.
intro nApp; destruct T; try solve [inversion 1]; cbn in *; eauto.
- intro H; apply andb_and in H. destruct H. rewrite H.
eapply leb_complete in H, H0. rewrite nth_error_inds; [lia|eauto].
- now destruct nApp.
Qed.
Lemma firstorder_type_closed_gen Σb kn l k T :
~ (isApp T) ->
firstorder_type (Σb := Σb) #|ind_bodies l| k T ->
closed (subst (inds kn [] (ind_bodies l)) k T).
Proof.
intro nApp; destruct T; try solve [inversion 1]; cbn in *; eauto.
- intro H; apply andb_and in H. destruct H. rewrite H.
eapply leb_complete in H, H0. rewrite nth_error_inds; [lia|eauto].
- now destruct nApp.