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cs cstr c : ConstraintSet.In c (ContextSet.constraints cs) -> ConstraintSet.In (uniquify_constraint_for (ContextSet.levels cs) true c) (ContextSet.constraints (declare_and_uniquify_and_combine_levels (cs, cstr)).1). Proof. cbv [declare_and_uniquify_levels declare_and_uniquify_and_combine_levels uniquify_constraint_for uniquify_constraint]. repeat first [ progress subst | progress cbn [ContextSet.constraints fst snd] | progress cbv [ConstraintSet.Exists] | destruct ? | rewrite ConstraintSet_In_fold_add | solve [ eauto ] ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | ConstraintSet_In__declare_and_uniquify_and_combine_levels_1__0 | 1,400 |
cs cstr c : ConstraintSet.In c (ContextSet.constraints (declare_and_uniquify_and_combine_levels (cs, cstr)).1) -> ConstraintSet.In (ununiquify_constraint 2 c) (ContextSet.constraints cs). Proof. cbv [declare_and_uniquify_levels declare_and_uniquify_and_combine_levels ununiquify_constraint uniquify_constraint_for uniquify_constraint]. repeat first [ progress subst | progress cbn [ContextSet.constraints fst snd] | progress cbv [ConstraintSet.Exists] | destruct ? | rewrite ConstraintSet_In_fold_add | rewrite ConstraintSetFact.empty_iff | progress intros | progress destruct_head'_and | progress destruct_head'_or | progress destruct_head'_ex | progress destruct_head'_False | rewrite ununiquify_level__uniquify_level by lia | match goal with | [ H : (_, _) = (_, _) |- _ ] => inv H end | solve [ eauto ] ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | ConstraintSet_In__declare_and_uniquify_and_combine_levels_1__1 | 1,401 |
cs cstr c : ConstraintSet.In c cstr -> ConstraintSet.In (uniquify_constraint_for (ContextSet.levels cs) false c) (declare_and_uniquify_and_combine_levels (cs, cstr)).2. Proof. cbv [declare_and_uniquify_levels declare_and_uniquify_and_combine_levels uniquify_constraint_for uniquify_constraint]. repeat first [ progress subst | progress cbn [ContextSet.constraints fst snd] | progress cbv [ConstraintSet.Exists] | destruct ? | rewrite ConstraintSet_In_fold_add | rewrite ConstraintSet.union_spec | solve [ eauto ] ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | ConstraintSet_In__declare_and_uniquify_and_combine_levels_2__0 | 1,402 |
cs cstr c : ConstraintSet.In c (declare_and_uniquify_levels (cs, cstr)).2 -> ConstraintSet.In (ununiquify_constraint 2 c) cstr. Proof. cbv [declare_and_uniquify_levels ununiquify_constraint uniquify_constraint_for uniquify_constraint]. repeat first [ progress subst | progress cbn [ContextSet.constraints fst snd] | progress cbv [ConstraintSet.Exists] | destruct ? | rewrite ConstraintSet_In_fold_add | rewrite ConstraintSetFact.empty_iff | progress intros | progress destruct_head'_and | progress destruct_head'_or | progress destruct_head'_ex | progress destruct_head'_False | rewrite ununiquify_level__uniquify_level by lia | match goal with | [ H : (_, _) = (_, _) |- _ ] => inv H end | solve [ eauto ] ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | ConstraintSet_In__declare_and_uniquify_levels_2__1 | 1,403 |
cs cstr side x : LevelSet.In x (ContextSet.levels cs) -> LevelSet.In (uniquify_level_for (ContextSet.levels cs) side x) (ContextSet.levels (declare_and_uniquify_and_combine_levels (cs, cstr)).1). Proof. cbv [declare_and_uniquify_and_combine_levels declare_and_uniquify_levels ContextSet.levels]; cbn [fst snd]. rewrite !LevelSet_In_fold_add. intro Hx. repeat lazymatch goal with | [ |- ?x \/ ?y ] => first [ lazymatch x with | context[LevelSet.Exists _ cs.1] => left end | lazymatch y with | context[LevelSet.Exists _ cs.1] => right end ] end. cbv [LevelSet.Exists uniquify_level_var uniquify_level_level uniquify_level_for uniquify_level]. exists x; split; trivial. destruct x; try reflexivity. all: now rewrite LevelSetFact.mem_1 by assumption. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | LevelSet_In_declare_and_uniquify_and_combine_levels_1_1 | 1,404 |
{cs cstr v} : satisfies v (ContextSet.constraints (declare_and_uniquify_and_combine_levels (cs, cstr)).1) -> satisfies (uniquify_valuation_for (ContextSet.levels cs) true v) (ContextSet.constraints cs). Proof. cbv [satisfies ConstraintSet.For_all uniquify_valuation_for]. intros H x Hi; specialize (H _ ltac:(eapply ConstraintSet_In__declare_and_uniquify_and_combine_levels_1__0, Hi)). destruct x as [[l []] r]; cbn in *; inversion H; clear H; subst; constructor. all: destruct l, r; assumption. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | satisfies_declare_and_uniquify_and_combine_levels_1_0 | 1,405 |
{cs cstr v} : satisfies v (ContextSet.constraints cs) -> satisfies (ununiquify_valuation 2 v) (ContextSet.constraints (declare_and_uniquify_and_combine_levels (cs, cstr)).1). Proof. cbv [satisfies ConstraintSet.For_all ununiquify_valuation]. intros H x Hi; specialize (H _ ltac:(eapply ConstraintSet_In__declare_and_uniquify_and_combine_levels_1__1, Hi)). destruct x as [[l []] r]; cbn in *; inversion H; clear H; subst; constructor. all: destruct l, r; assumption. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | satisfies_declare_and_uniquify_and_combine_levels_1_1 | 1,406 |
{cs cstr v} : satisfies v (declare_and_uniquify_and_combine_levels (cs, cstr)).2 -> satisfies (uniquify_valuation_for (ContextSet.levels cs) false v) cstr. Proof. cbv [satisfies ConstraintSet.For_all uniquify_valuation_for]. intros H x Hi; specialize (H _ ltac:(eapply ConstraintSet_In__declare_and_uniquify_and_combine_levels_2__0, Hi)). destruct x as [[l []] r]; cbn in *; inversion H; clear H; subst; constructor. all: destruct l, r; assumption. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | satisfies_declare_and_uniquify_and_combine_levels_2_0 | 1,407 |
{cs cstr v} : satisfies v cstr -> satisfies (ununiquify_valuation 2 v) (declare_and_uniquify_levels (cs, cstr)).2. Proof. cbv [satisfies ConstraintSet.For_all uniquify_valuation_for]. intros H x Hi; specialize (H _ ltac:(eapply ConstraintSet_In__declare_and_uniquify_levels_2__1, Hi)). destruct x as [[l []] r]; cbn in *; inversion H; clear H; subst; constructor. all: destruct l, r; try assumption. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | satisfies_declare_and_uniquify_levels_2_1 | 1,408 |
{cs cstr v v'} (cscstr := declare_and_uniquify_levels (cs, cstr)) (cscstr' := declare_and_uniquify_and_combine_levels (cs, cstr)) (cs' := cscstr'.1) (cstr' := cscstr.2) (cstr'' := cscstr'.2) (Hv : satisfies v (ContextSet.constraints cs')) (Hv' : satisfies v' cstr') (Hagree : LevelSet.For_all (fun l => val v (uniquify_level_for (ContextSet.levels cs) true l) = val v' (uniquify_level_for (ContextSet.levels cs) false l)) (ContextSet.levels cs)) (vc := combine_valuations "b"%byte "l"%byte "r"%byte v v v') : satisfies vc cstr'' /\ LevelSet.For_all (fun l => val v l = val vc l) (ContextSet.levels cs'). Proof. repeat match goal with H := _ |- _ => subst H end. cbv [satisfies ConstraintSet.For_all LevelSet.For_all combine_valuations val Level.Evaluable ContextSet.constraints ContextSet.levels declare_and_uniquify_and_combine_levels declare_and_uniquify_levels] in *; cbn [fst snd valuation_poly valuation_mono] in *. revert Hv Hv' Hagree. progress repeat setoid_rewrite ConstraintSet.union_spec. progress repeat setoid_rewrite LevelSet_In_fold_add. progress repeat setoid_rewrite ConstraintSet_In_fold_add. progress repeat setoid_rewrite ConstraintSetFact.empty_iff. progress repeat setoid_rewrite LevelSet.singleton_spec. cbv [LevelSet.Exists ConstraintSet.Exists uniquify_constraint_for uniquify_constraint uniquify_level_for uniquify_level]. intros. split. 2: intro x; specialize (Hagree (ununiquify_level 2 x)). 2: cbv [ununiquify_level ununiquify_level_level ununiquify_level_var] in *. all: repeat first [ progress intros | progress subst | progress rdest | progress destruct_head'_False | progress destruct_head'_or | progress destruct_head'_ex | progress specialize_by_assumption | progress cbv beta iota in * | reflexivity | match goal with | [ H : forall x, _ \/ _ -> _ |- _ ] => pose proof (fun x H' => H x (or_introl H')); pose proof (fun x H' => H x (or_intror H')); clear H | [ H : _ \/ _ -> _ |- _ ] => pose proof (fun H' => H (or_introl H')); pose proof (fun H' => H (or_intror H')); clear H | [ H : forall x, ex _ -> _ |- _ ] => specialize (fun x x' H' => H x (ex_intro _ x' H')) | [ H : ex _ -> _ |- _ ] => specialize (fun x' H' => H (ex_intro _ x' H')) | [ H : forall x x', _ /\ x = @?f x' -> _ |- _ ] => specialize (fun x' H' => H _ x' (conj H' eq_refl)) | [ H : forall x, _ /\ _ = _ -> _ |- _ ] => specialize (fun H' => H _ (conj H' eq_refl)) | [ H : forall x, x = _ -> _ |- _ ] => specialize (H _ eq_refl) | [ H : forall x, False -> _ |- _ ] => clear H end ]. all: repeat first [ progress cbv [uniquify_level_level uniquify_level_var] in * | congruence | lia | progress subst | match goal with | [ H : Level.lvar _ = Level.lvar _ |- _ ] => inversion H; clear H | [ H : Level.level _ = Level.level _ |- _ ] => inversion H; clear H | [ H : (@eqb ?T ?R ?x ?y) = true |- _ ] => destruct (@eqb_spec T R x y) | [ H : (@eqb ?T ?R ?x ?y) = false |- _ ] => destruct (@eqb_spec T R x y) end | progress destruct ? ]. all: repeat first [ progress rewrite ?Nat2Z.inj_add, ?Nat2Z.inj_mul in * | progress change (Z.of_nat 3) with 3%Z in * | progress change (?n mod 3)%Z with n in * | match goal with | [ H : context[((?x * ?y + ?z) mod ?y)%Z] |- _ ] => rewrite (Z.add_comm (x * y) z) in * end | progress rewrite ?Z_mod_plus_full in * | lia ]. all: repeat match goal with | [ H : LevelSet.In _ _ |- _ ] => progress specialize_all_ways_under_binders_by exact H | [ H : ConstraintSet.In _ _ |- _ ] => progress specialize_all_ways_under_binders_by exact H end. all: repeat first [ progress subst | assumption | progress cbv [val Level.Evaluable] in * | progress cbn [fst snd valuation_mono valuation_poly] in * | progress destruct_head_hnf' prod | match goal with | [ H : satisfies0 _ _ |- _ ] => inversion H; clear H; constructor end ]. all: repeat first [ progress cbv [uniquify_level_level uniquify_level_var] in * | rewrite eqb_refl | assumption | match goal with | [ H : ?x = true, H' : context[match ?x with _ => _ end] |- _ ] => rewrite H in H' | [ H : ?x = false, H' : context[match ?x with _ => _ end] |- _ ] => rewrite H in H' | [ |- context[LevelSet.mem ?l ?x] ] => let H := fresh in pose proof (@LevelSetFact.mem_2 x l) as H; destruct (LevelSet.mem l x) eqn:?; try (specialize (H eq_refl); specialize_all_ways_under_binders_by exact H) | [ H : LevelSet.mem ?x ?l = true |- _ ] => unique pose proof (@LevelSetFact.mem_2 _ _ H); let H' := match goal with H' : LevelSet.In x l |- _ => H' end in specialize_all_ways_under_binders_by exact H' end ]. all: repeat first [ progress cbv beta iota in * | rewrite !Nat2Z.inj_add, !Nat2Z.inj_mul | progress change (Z.of_nat 3) with 3%Z | rewrite Z.add_comm, Z_mod_plus_full | rewrite eqb_refl | assumption | lia | match goal with | [ |- context[(?x == ?y)] ] => change (x == y) with false end ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | satisfies_combine_valuations | 1,409 |
cs cstr : @consistent_extension_on cs cstr <-> @consistent_extension_on (declare_and_uniquify_and_combine_levels (cs, cstr)).1 (declare_and_uniquify_and_combine_levels (cs, cstr)).2. Proof. cbv [consistent_extension_on]. split; intros H v Hs. { specialize (H _ (satisfies_declare_and_uniquify_and_combine_levels_1_0 Hs)). destruct H as [v' [H0 H1]]. apply (@satisfies_declare_and_uniquify_levels_2_1 cs cstr) in H0. eexists; eapply satisfies_combine_valuations; try eassumption. revert H1. cbv [LevelSet.For_all ununiquify_valuation uniquify_valuation_for uniquify_valuation val Level.Evaluable uniquify_level_for uniquify_level]. cbn [valuation_mono valuation_poly]. intros H1 x Hx; specialize (H1 x Hx); revert H1. destruct x; try lia. all: first [ rewrite ununiquify_level_level__uniquify_level_level | rewrite ununiquify_level_var__uniquify_level_var by lia ]. all: trivial. } { specialize (H _ (satisfies_declare_and_uniquify_and_combine_levels_1_1 Hs)). destruct H as [v' [H0 H1]]. eexists; split; [ eapply satisfies_declare_and_uniquify_and_combine_levels_2_0; eassumption | ]. cbv [LevelSet.For_all] in *. intros l Hl; specialize (fun side => H1 _ ltac:(unshelve eapply LevelSet_In_declare_and_uniquify_and_combine_levels_1_1, Hl; exact side)). pose proof (H1 true) as H1t. pose proof (H1 false) as H1f. clear H1. cbv [val Level.Evaluable ununiquify_valuation uniquify_level_for uniquify_level uniquify_valuation_for uniquify_valuation] in *. destruct l; trivial. cbn [valuation_poly valuation_mono] in *. rewrite ?ununiquify_level_var__uniquify_level_var in * by lia. congruence. } Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | consistent_extension_on_iff_declare_and_uniquify_and_combine_levels | 1,410 |
cs cstr : global_uctx_invariants (declare_and_uniquify_and_combine_levels (cs, cstr)).1. Proof. pose proof (levels_of_cs_spec (ContextSet.constraints cs)). pose proof (levels_of_cs_spec cstr). cbv [declare_and_uniquify_levels]; cbn [fst snd]. cbv [uGraph.global_uctx_invariants uGraph.uctx_invariants ConstraintSet.For_all declared_cstr_levels] in *; cbn [fst snd ContextSet.levels ContextSet.constraints] in *. repeat first [ progress subst | progress cbv [LevelSet.Exists ConstraintSet.Exists uniquify_constraint_for uniquify_constraint uniquify_level_for] in * | rewrite !LevelSet_In_fold_add | rewrite !ConstraintSet_In_fold_add | rewrite !LevelSet.singleton_spec | rewrite ConstraintSetFact.empty_iff | setoid_rewrite LevelSet_In_fold_add | setoid_rewrite ConstraintSet_In_fold_add | setoid_rewrite LevelSet.singleton_spec | setoid_rewrite ConstraintSetFact.empty_iff | match goal with | [ H : (_, _) = (_, _) |- _ ] => inv H | [ H : forall x : ConstraintSet.elt, _ |- _ ] => specialize (fun a b c => H ((a, b), c)) end | solve [ eauto ] | progress rdest | progress destruct_head'_ex | progress split_and | progress intros | progress destruct ? | progress destruct_head'_or ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | global_uctx_invariants__declare_and_uniquify_and_combine_levels | 1,411 |
cs cstr G G' (cscstr := declare_and_uniquify_and_combine_levels (cs, cstr)) (cs' := cscstr.1) (cstr' := cscstr.2) (cf := config.default_checker_flags) (lvls := levels_of_cscs cs' cstr') (HG : gc_of_uctx cs' = Some G) (HG' : gc_of_uctx (lvls, cstr') = Some G') : subgraph (make_graph G) (make_graph G'). Proof. repeat first [ progress cbv [gc_of_uctx monad_utils.bind monad_utils.ret monad_utils.option_monad] in * | progress cbn [fst snd] in * | progress subst | progress destruct ? | match goal with | [ H : Some ?x = Some ?y |- _ ] => assert (x = y) by congruence; clear H end | congruence ]. repeat match goal with H := _ |- _ => subst H end. split; try reflexivity; cbv [levels_of_cscs ContextSet.levels uGraph.wGraph.E make_graph uGraph.wGraph.V]; cbn [fst snd] in *; try solve [ clear; LevelSetDecide.fsetdec ]; []. all: lazymatch goal with | [ |- EdgeSet.Subset _ _ ] => idtac end. intro; rewrite !add_cstrs_spec, !add_level_edges_spec, !EdgeSetFact.empty_iff; repeat setoid_rewrite VSet.union_spec. all: repeat first [ intro | progress destruct_head'_or | progress destruct_head'_ex | progress destruct_head'_and | progress subst | exfalso; assumption | progress rewrite ?@gc_of_constraint_iff in * by eassumption | progress cbv [ConstraintSet.Exists on_Some] in * | progress destruct ? | solve [ eauto 6 ] ]. all: [ > ]. left; eexists; split; [ reflexivity | ]. all: repeat first [ intro | progress destruct_head'_or | progress destruct_head'_ex | progress destruct_head'_and | progress subst | exfalso; assumption | progress rewrite ?@gc_of_constraint_iff in * by eassumption | progress cbv [ConstraintSet.Exists on_Some] in * | progress destruct ? | solve [ eauto 6 ] ]. eexists; split; [ | match goal with H : _ |- _ => rewrite H; eassumption end ]. cbv [declare_and_uniquify_and_combine_levels ContextSet.constraints] in *; cbn [fst snd] in *. ConstraintSetDecide.fsetdec. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | consistent_extension_on_iff_subgraph_helper | 1,412 |
cs cstr (cscstr := declare_and_uniquify_and_combine_levels (cs, cstr)) (cs' := cscstr.1) (cstr' := cscstr.2) (cf := config.default_checker_flags) (lvls := levels_of_cscs cs' cstr') : @consistent_extension_on cs cstr <-> is_true match uGraph.is_consistent cs', uGraph.is_consistent (lvls, cstr'), uGraph.gc_of_uctx cs', uGraph.gc_of_uctx (lvls, cstr') with | false, _, _, _ | _, _, None, _ => true | _, true, Some G, Some G' => uGraph.wGraph.IsFullSubgraph.is_full_extension (uGraph.make_graph G) (uGraph.make_graph G') | _, _, _, _ => false end. Proof. rewrite consistent_extension_on_iff_declare_and_uniquify_and_combine_levels. destruct (levels_of_cscs_spec cs' cstr'). subst cscstr cs' cstr'. cbv zeta; repeat destruct ?; subst. let H := fresh in pose proof (fun uctx uctx' G => @uGraph.consistent_ext_on_full_ext _ uctx G (lvls, uctx')) as H; cbn [fst snd] in H; erewrite H; clear H. 1: reflexivity. all: cbn [fst snd ContextSet.constraints] in *. all: repeat repeat first [ match goal with | [ H : _ = Some _ |- _ ] => rewrite H | [ H : _ = None |- _ ] => rewrite H | [ |- _ <-> is_true false ] => cbv [is_true]; split; [ let H := fresh in intro H; contradict H | congruence ] | [ |- _ <-> is_true true ] => split; [ reflexivity | intros _ ] end | progress cbv [uGraph.is_graph_of_uctx monad_utils.bind monad_utils.ret monad_utils.option_monad] in * | progress cbn [MCOption.on_Some fst snd] in * | rewrite <- uGraph.is_consistent_spec2 | progress subst | assert_fails (idtac; lazymatch goal with |- ?G => has_evar G end); first [ reflexivity | assumption ] | match goal with | [ H : ?T, H' : ~?T |- _ ] => exfalso; apply H', H | [ H : context[match ?x with _ => _ end] |- _ ] => destruct x eqn:? | [ H : uGraph.gc_of_uctx _ = None |- _ ] => cbv [uGraph.gc_of_uctx] in * | [ H : Some _ = Some _ |- _ ] => inversion H; clear H | [ H : Some _ = None |- _ ] => inversion H | [ H : None = Some _ |- _ ] => inversion H | [ H : ?T <-> False |- _ ] => destruct H as [H _]; try change (~T) in H | [ H : ~consistent ?cs |- consistent_extension_on (_, ?cs) _ ] => intros ? ?; exfalso; apply H; eexists; eassumption | [ H : ~consistent (snd ?cs) |- consistent_extension_on ?cs _ ] => intros ? ?; exfalso; apply H; eexists; eassumption | [ H : @uGraph.is_consistent ?cf ?uctx = false |- _ ] => assert (~consistent (snd uctx)); [ rewrite <- (@uGraph.is_consistent_spec cf uctx), H; clear H; auto | clear H ] | [ H : @uGraph.gc_of_constraints ?cf ?ctrs = None |- _ ] => let H' := fresh in pose proof (@uGraph.gc_consistent_iff cf ctrs) as H'; rewrite H in H'; clear H | [ H : @uGraph.is_consistent ?cf ?uctx = true |- _ ] => assert_fails (idtac; match goal with | [ H' : consistent ?v |- _ ] => unify v (snd uctx) end); assert (consistent (snd uctx)); [ rewrite <- (@uGraph.is_consistent_spec cf uctx), H; clear H; auto | ] end ]. all: try now apply global_uctx_invariants__declare_and_uniquify_and_combine_levels. all: try now eapply @consistent_extension_on_iff_subgraph_helper. all: try solve [ repeat first [ progress cbv [consistent consistent_extension_on not] in * | progress intros | progress destruct_head'_ex | progress destruct_head'_and | progress specialize_under_binders_by eassumption | solve [ eauto ] ] ]. Qed. | Lemma | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | consistent_extension_on_iff | 1,413 |
cs cstr : {@consistent_extension_on cs cstr} + {~@consistent_extension_on cs cstr}. Proof. pose proof (@consistent_extension_on_iff cs cstr) as H; cbv beta zeta in *. let b := lazymatch type of H with context[is_true ?b] => b end in destruct b; [ left; apply H; reflexivity | right; intro H'; apply H in H'; auto ]. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | consistent_extension_on_dec | 1,414 |
n ϕ u u' : {@leq0_universe_n (uGraph.Z_of_bool n) ϕ u u'} + {~@leq0_universe_n (uGraph.Z_of_bool n) ϕ u u'}. Proof. pose proof (@uGraph.gc_leq0_universe_n_iff config.default_checker_flags (uGraph.Z_of_bool n) ϕ u u') as H. pose proof (@uGraph.gc_consistent_iff config.default_checker_flags ϕ). cbv [on_Some on_Some_or_None] in *. destruct gc_of_constraints eqn:?. all: try solve [ left; cbv [consistent] in *; hnf; intros; exfalso; intuition eauto ]. pose proof (fun G cstr => @uGraph.leqb_universe_n_spec G (LevelSet.union (levels_of_cs ϕ) (LevelSet.union (levels_of_universe u) (levels_of_universe u')), cstr)). pose proof (fun x y => @gc_of_constraints_of_uctx config.default_checker_flags (x, y)) as H'. pose proof (@is_consistent_spec config.default_checker_flags (levels_of_cs ϕ, ϕ)). specialize_under_binders_by eapply gc_levels_declared_union_or. specialize_under_binders_by eapply global_gc_uctx_invariants_union_or. specialize_under_binders_by (constructor; eapply gc_of_uctx_invariants). cbn [fst snd] in *. specialize_under_binders_by eapply H'. specialize_under_binders_by eassumption. specialize_under_binders_by eapply levels_of_cs_spec. specialize_under_binders_by reflexivity. destruct is_consistent; [ | left; now cbv [leq0_universe_n consistent] in *; intros; exfalso; intuition eauto ]. specialize_by intuition eauto. let H := match goal with H : forall (b : bool), _ |- _ => H end in specialize (H n u u'). specialize_under_binders_by (constructor; eapply gc_levels_declared_union_or; constructor; eapply levels_of_universe_spec). match goal with H : is_true ?b <-> ?x, H' : ?y <-> ?x |- {?y} + {_} => destruct b eqn:?; [ left | right ] end. all: intuition. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | leq0_universe_n_dec | 1,415 |
cf n ϕ u u' : {@leq_universe_n cf (uGraph.Z_of_bool n) ϕ u u'} + {~@leq_universe_n cf (uGraph.Z_of_bool n) ϕ u u'}. Proof. cbv [leq_universe_n]; destruct (@leq0_universe_n_dec n ϕ u u'); destruct ?; auto. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | leq_universe_n_dec | 1,416 |
ϕ u u' : {@eq0_universe ϕ u u'} + {~@eq0_universe ϕ u u'}. Proof. pose proof (@eq0_leq0_universe ϕ u u') as H. destruct (@leq0_universe_n_dec false ϕ u u'), (@leq0_universe_n_dec false ϕ u' u); constructor; destruct H; split_and; now auto. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | eq0_universe_dec | 1,417 |
{cf ϕ} u u' : {@eq_universe cf ϕ u u'} + {~@eq_universe cf ϕ u u'}. Proof. cbv [eq_universe]; destruct ?; auto using eq0_universe_dec. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | eq_universe_dec | 1,418 |
{univ eq_universe} (eq_universe_dec : forall u u', {@eq_universe u u'} + {~@eq_universe u u'}) s s' : {@eq_sort_ univ eq_universe s s'} + {~@eq_sort_ univ eq_universe s s'}. Proof. cbv [eq_sort_]; repeat destruct ?; auto. all: destruct pst; auto. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | eq_sort__dec | 1,419 |
{cf ϕ} s s' : {@eq_sort cf ϕ s s'} + {~@eq_sort cf ϕ s s'} := eq_sort__dec eq_universe_dec _ _. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | eq_sort_dec | 1,420 |
cf ϕ cstrs : {@valid_constraints cf ϕ cstrs} + {~@valid_constraints cf ϕ cstrs}. Proof. pose proof (fun G a b c => uGraph.check_constraints_spec (uGraph.make_graph G) (levels_of_cs2 ϕ cstrs, ϕ) a b c cstrs) as H1. pose proof (fun G a b c => uGraph.check_constraints_complete (uGraph.make_graph G) (levels_of_cs2 ϕ cstrs, ϕ) a b c cstrs) as H2. pose proof (levels_of_cs2_spec ϕ cstrs). cbn [fst snd] in *. destruct (consistent_dec ϕ); [ | now left; cbv [valid_constraints valid_constraints0 consistent not] in *; destruct ?; intros; eauto; exfalso; eauto ]. destruct_head'_and. specialize_under_binders_by assumption. cbv [uGraph.is_graph_of_uctx MCOption.on_Some] in *. cbv [valid_constraints] in *; repeat destruct ?; auto. { specialize_under_binders_by reflexivity. destruct uGraph.check_constraints_gen; specialize_by reflexivity; auto. } { rewrite uGraph.gc_consistent_iff in *. cbv [uGraph.gc_of_uctx monad_utils.bind monad_utils.ret monad_utils.option_monad MCOption.on_Some] in *; cbn [fst snd] in *. destruct ?. all: try congruence. all: exfalso; assumption. } Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | valid_constraints_dec | 1,421 |
ϕ ctrs : {@valid_constraints0 ϕ ctrs} + {~@valid_constraints0 ϕ ctrs} := @valid_constraints_dec config.default_checker_flags ϕ ctrs. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | valid_constraints0_dec | 1,422 |
cf ϕ s : {@is_lSet cf ϕ s} + {~@is_lSet cf ϕ s}. Proof. apply eq_sort_dec. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | is_lSet_dec | 1,423 |
cf ϕ allowed u : {@is_allowed_elimination cf ϕ allowed u} + {~@is_allowed_elimination cf ϕ allowed u}. Proof. cbv [is_allowed_elimination is_true]; destruct ?; auto; try solve [ repeat decide equality ]. destruct (@is_lSet_dec cf ϕ u), Sort.is_propositional; intuition auto. Defined. | Definition | common | From Coq Require Import PArith NArith ZArith Lia. From MetaCoq.Utils Require Import MCList MCOption MCUtils. From MetaCoq.Common Require Import uGraph. From MetaCoq.Common Require Import Universes. Import wGraph. | common\theories\UniversesDec.v | is_allowed_elimination_dec | 1,424 |
Σ Γ t : Is_proof Σ Γ t -> isErasable Σ Γ t. Proof. intros. destruct X as (? & ? & ? & ? & ?). exists x. split. eauto. right. eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isErasable_Proof | 1,425 |
Σ Γ T : isType Σ Γ T -> isErasable Σ Γ T. Proof. intros (_ & s & Hs & _). exists (tSort s). intuition auto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isType_isErasable | 1,426 |
forall (Σ : global_env_ext) (Γ : context) (T : term), wf Σ -> wf_local Σ Γ -> isType Σ Γ T -> forall T' : term, red Σ Γ T T' -> isType Σ Γ T'. Proof. intros. apply lift_sorting_it_impl_gen with X1 => // HT. eapply subject_reduction ; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isType_red | 1,427 |
forall (l : list context_decl) A, isArity A -> isArity (it_mkProd_or_LetIn l A). Proof. induction l; cbn; intros; eauto. eapply IHl. destruct a, decl_body; cbn; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | it_mkProd_isArity | 1,428 |
(Σ : global_env_ext) mind ind idecl : wf Σ -> declared_inductive (fst Σ) ind mind idecl -> isArity (ind_type idecl). Proof. intros. eapply (declared_inductive_inv weaken_env_prop_typing) in H; eauto. - inv H. rewrite ind_arity_eq. change PCUICEnvironment.it_mkProd_or_LetIn with it_mkProd_or_LetIn. rewrite <- it_mkProd_or_LetIn_app. clear. eapply it_mkProd_isArity. econstructor. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isArity_ind_type | 1,429 |
(Σ : global_env_ext) (Γ : context) (x : aname) (x0 x1 : term) : wf Σ -> isWfArity Σ Γ (tProd x x0 x1) -> (isType Σ Γ x0 × isWfArity Σ (Γ,, vass x x0) x1). Proof. intros wfΣ (? & ? & ? & ?). cbn in e. eapply isType_tProd in i as [dom codom]; auto. split; auto. split; auto. clear dom codom. eapply destArity_app_Some in e as (? & ? & ?); subst. eexists. eexists; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isWfArity_prod_inv | 1,430 |
ind u l n t : nth_error (inds ind u l) n = Some t -> exists n, t = tInd {| inductive_mind := ind ; inductive_ind := n |} u. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | inds_nth_error | 1,431 |
forall (l : list context_decl) (A : term), isArity (it_mkProd_or_LetIn l A) -> isArity A. Proof. induction l; cbn; intros. - eauto. - eapply IHl in H. destruct a, decl_body; cbn in *; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | it_mkProd_arity | 1,432 |
t L : isArity (mkApps t L) -> isArity t /\ L = []. Proof. revert t; induction L; cbn; intros. - eauto. - eapply IHL in H. cbn in H. intuition. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isArity_mkApps | 1,433 |
(Σ : global_env_ext) Γ (args args' : list PCUICAst.term) (X : All2 (red Σ Γ) args args') (wfΣ : wf Σ) (T x x0 : PCUICAst.term) (t0 : typing_spine Σ Γ x args x0) (c : Σ;;; Γ ⊢ x0 ≤ T) x1 (c0 : Σ;;; Γ ⊢ x1 ≤ x) : isType Σ Γ x1 -> isType Σ Γ T -> typing_spine Σ Γ x1 args' T. Proof. intros ? ?. revert args' X. dependent induction t0; intros. - inv X. econstructor; eauto. transitivity ty => //. now transitivity ty'. - inv X. econstructor; tea. + transitivity ty => //. + eapply subject_reduction; eauto. + eapply IHt0; eauto. eapply red_ws_cumul_pb_inv. unfold subst1. eapply isType_tProd in i0 as [dom codom]. eapply (closed_red_red_subst (Δ := [vass na A]) (Γ' := [])); auto. simpl. eapply isType_wf_local in codom. fvs. constructor; auto. eapply into_closed_red; auto. fvs. fvs. repeat constructor. eapply isType_is_open_term in codom; fvs. eapply isType_apply in i0; tea. eapply subject_reduction; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | typing_spine_red | 1,434 |
{Σ : global_env_ext} {Γ c0 i u l} {wfΣ : wf Σ} : ~ Is_conv_to_Arity Σ Γ (it_mkProd_or_LetIn c0 (mkApps (tInd i u) l)). Proof. intros (? & [] & ?). eapply red_it_mkProd_or_LetIn_mkApps_Ind in X as (? & ? & ?). subst. eapply it_mkProd_arity in H. eapply isArity_mkApps in H as [? ?]. cbn in *. congruence. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | it_mkProd_red_Arity | 1,435 |
forall (u : Instance.t) (i : inductive) (x : aname) (x0 x1 : term) (x2 : context) (x3 : list term), tProd x x0 x1 = it_mkProd_or_LetIn x2 (mkApps (tInd i u) x3) -> exists (L' : context) (l' : list term), x1 = it_mkProd_or_LetIn L' (mkApps (tInd i u) l'). Proof. intros u i x x0 x1 x2 x3 H0. revert x0 x3 x1 x H0. induction x2 using rev_ind; intros. - cbn. assert (decompose_app (tProd x x0 x1) = decompose_app (mkApps (tInd i u) x3)) by now rewrite H0. rewrite decompose_app_mkApps in H; cbn; eauto. cbn in H. inv H. - rewrite it_mkProd_or_LetIn_app in H0. cbn in *. destruct x, decl_body; cbn in H0; try now inv H0. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | invert_it_Ind_eq_prod | 1,436 |
{Σ : global_env_ext} {wfΣ : wf Σ} {Γ} {ind n mdecl idecl cdecl u} : declared_constructor Σ (ind, n) mdecl idecl cdecl -> ~ Is_conv_to_Arity Σ Γ (type_of_constructor mdecl cdecl (ind, n) u). Proof. intros decl; sq. unfold type_of_constructor. destruct (on_declared_constructor decl) as [XX [s [XX1 Ht]]]. rewrite (cstr_eq Ht). clear -wfΣ decl. rewrite !PCUICUnivSubst.subst_instance_it_mkProd_or_LetIn !subst_it_mkProd_or_LetIn. rewrite /cstr_concl. rewrite /cstr_concl_head. len. rewrite subst_cstr_concl_head. destruct decl as [[] ?]. now eapply nth_error_Some_length in H0. rewrite -it_mkProd_or_LetIn_app. now eapply it_mkProd_red_Arity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | declared_constructor_type_not_arity | 1,437 |
{Σ : global_env_ext} {wfΣ : wf Σ} : forall (Γ : context) (C : term) T, Is_conv_to_Arity Σ Γ T -> Σ;;; Γ ⊢ C ≤ T -> Is_conv_to_Arity Σ Γ C. Proof. intros Γ C T [? []] cum. sq. eapply invert_cumul_arity_r_gen; tea. exists x. split; auto. now sq. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | conv_to_arity_cumul | 1,438 |
{Σ : global_env_ext} {wfΣ : wf Σ} Γ Δ ind u args args' T' : typing_spine Σ Γ (it_mkProd_or_LetIn Δ (mkApps (tInd ind u) args)) args' T' -> ∑ Δ' args', Σ ;;; Γ ⊢ it_mkProd_or_LetIn Δ' (mkApps (tInd ind u) args') ≤ T'. Proof. induction Δ in args, args' |- * using PCUICInduction.ctx_length_rev_ind. - simpl. intros sp. dependent elimination sp as [spnil i i' e|spcons i i' e e' c]. * now exists [], args. * now eapply invert_cumul_ind_prod in e. - rewrite it_mkProd_or_LetIn_app /=; destruct d as [na [b|] ty]. * rewrite /mkProd_or_LetIn /=. simpl => /= sp. eapply typing_spine_letin_inv in sp; eauto. rewrite /subst1 subst_it_mkProd_or_LetIn Nat.add_0_r subst_mkApps /= in sp. apply (X (subst_context [b] 0 Γ0) ltac:(now len) _ _ sp). * rewrite /mkProd_or_LetIn /=. simpl => /= sp. simpl. dependent elimination sp as [spnil i i' e|spcons i i' e e' sp]. { exists (Γ0 ++ [vass na ty]). exists args. now rewrite it_mkProd_or_LetIn_app. } eapply ws_cumul_pb_Prod_Prod_inv in e as [eqna dom codom]; pcuic. eapply (substitution0_ws_cumul_pb (t:=hd0)) in codom; eauto. eapply typing_spine_strengthen in sp. 3:tea. rewrite /subst1 subst_it_mkProd_or_LetIn Nat.add_0_r subst_mkApps /= in sp. apply (X (subst_context [hd0] 0 Γ0) ltac:(len; reflexivity) _ _ sp). eapply isType_apply in i; tea. eapply (type_ws_cumul_pb (pb:=Conv)); tea. 2:now symmetry. now eapply isType_tProd in i as []. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | typing_spine_mkApps_Ind_ex | 1,439 |
{Σ : global_env_ext} {wfΣ : wf Σ} {Γ Δ ind u args args' T'} : typing_spine Σ Γ (it_mkProd_or_LetIn Δ (mkApps (tInd ind u) args)) args' T' -> ~ Is_conv_to_Arity Σ Γ T'. Proof. move/typing_spine_mkApps_Ind_ex => [Δ' [args'' cum]]. intros iscv. eapply invert_cumul_arity_r_gen in iscv; tea. now eapply it_mkProd_red_Arity in iscv. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | typing_spine_Is_conv_to_Arity | 1,440 |
{Σ : global_env_ext} {wfΣ : wf Σ} {Γ} {ind n mdecl idecl cdecl u args' T'} : declared_constructor Σ (ind, n) mdecl idecl cdecl -> typing_spine Σ Γ (type_of_constructor mdecl cdecl (ind, n) u) args' T' -> ~ Is_conv_to_Arity Σ Γ T'. Proof. intros decl; sq. unfold type_of_constructor. destruct (on_declared_constructor decl) as [XX [s [XX1 Ht]]]. rewrite (cstr_eq Ht). clear -wfΣ decl. rewrite !PCUICUnivSubst.subst_instance_it_mkProd_or_LetIn !subst_it_mkProd_or_LetIn. rewrite /cstr_concl. rewrite /cstr_concl_head. len. rewrite subst_cstr_concl_head. destruct decl as [[] ?]. now eapply nth_error_Some_length in H0. rewrite -it_mkProd_or_LetIn_app. apply typing_spine_Is_conv_to_Arity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | declared_constructor_typing_spine_not_arity | 1,441 |
(Σ : global_env_ext) Γ ind c u x1 T : wf Σ -> Σ ;;; Γ |- mkApps (tConstruct ind c u) x1 : T -> ~ Is_conv_to_Arity Σ Γ T. Proof. intros. eapply PCUICValidity.inversion_mkApps in X0 as (? & ? & ?); eauto. eapply inversion_Construct in t as (? & ? & ? & ? & ? & ? & ?) ; auto. eapply typing_spine_strengthen in t0. 3:tea. eapply declared_constructor_typing_spine_not_arity in t0; tea. eapply PCUICValidity.validity. econstructor; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | type_mkApps_tConstruct_n_conv_arity | 1,442 |
{Σ : global_env_ext} {wfΣ : wf Σ} {Γ T} : isType Σ Γ T -> ~ Is_conv_to_Arity Σ Γ T -> ~ isArity T. Proof. intros isty nisc isa. apply nisc. exists T. split => //. sq. destruct isty as (_ & s & Hs & _). eapply wt_closed_red_refl; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | nIs_conv_to_Arity_nArity | 1,443 |
(Σ : global_env_ext) Γ ind c u x1 : wf Σ -> isErasable Σ Γ (mkApps (tConstruct ind c u) x1) -> Is_proof Σ Γ (mkApps (tConstruct ind c u) x1). Proof. intros wfΣ (? & ? & [ | (? & ? & ?)]). - exfalso. eapply nIs_conv_to_Arity_nArity; tea. eapply PCUICValidity.validity; tea. eapply type_mkApps_tConstruct_n_conv_arity in t; auto. - exists x, x0. eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | tConstruct_no_Type | 1,444 |
(Σ : global_env_ext) Γ mfix idx x1 : wf Σ -> isErasable Σ Γ (mkApps (tCoFix mfix idx) x1) -> Is_proof Σ Γ (mkApps (tCoFix mfix idx) x1). Proof. intros wfΣ (? & ? & [ | (? & ? & ?)]). - exfalso. eapply PCUICValidity.inversion_mkApps in t as (? & ? & ?); eauto. pose proof (typing_spine_isType_codom t0). assert(c0 : Σ ;;; Γ ⊢ x ≤ x) by now eapply (isType_ws_cumul_pb_refl). revert c0 t0 i. generalize x at 1 3. intros x2 c0 t0 i. assert (HWF : isType Σ Γ x2). { eapply PCUICValidity.validity. eapply type_mkApps. 2:eauto. eauto. } eapply inversion_CoFix in t as (? & ? & ? & ? & ? & ? & ?) ; auto. eapply invert_cumul_arity_r in c0; eauto. eapply typing_spine_strengthen in t0. 3:eauto. eapply wf_cofixpoint_spine in i0; eauto. 2-3:eapply nth_error_all in a; eauto; simpl in a; eauto. destruct i0 as (Γ' & T & DA & ind & u & indargs & (eqT & ck) & cum). destruct (Nat.ltb #|x1| (context_assumptions Γ')). eapply invert_cumul_arity_r_gen in c0; eauto. destruct c0. destruct H as [[r] isA]. move: r; rewrite subst_it_mkProd_or_LetIn eqT; autorewrite with len. rewrite PCUICSigmaCalculus.expand_lets_mkApps subst_mkApps /=. move/red_it_mkProd_or_LetIn_mkApps_Ind => [ctx' [args' eq]]. subst x4. now eapply it_mkProd_arity, isArity_mkApps in isA. move: cum => [] Hx1; rewrite eqT PCUICSigmaCalculus.expand_lets_mkApps subst_mkApps /= => cum. eapply invert_cumul_arity_r_gen in c0; eauto. now eapply Is_conv_to_Arity_ind in c0. - eexists _, _; intuition eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | tCoFix_no_Type | 1,445 |
(Σ : global_env_ext) (Γ : context) (L : list term) (x x0 : term) : wf Σ -> typing_spine Σ Γ x L x0 -> isType Σ Γ x0. Proof. intros wfΣ; induction 1; auto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | typing_spine_wat | 1,446 |
Γ A B u : Sort.is_prop u -> isType Σ Γ A -> Σ ;;; Γ |- B : tSort u -> Σ ;;; Γ ⊢ A ≤ B -> Σ ;;; Γ |- A : tSort u. Proof using Hcf wfΣ. intros; eapply cumul_prop1; tea. now apply ws_cumul_pb_forget in X1. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_prop1 | 1,447 |
Γ A B u : Sort.is_prop u -> isType Σ Γ B -> Σ ;;; Γ ⊢ A ≤ B -> Σ ;;; Γ |- A : tSort u -> Σ ;;; Γ |- B : tSort u. Proof using Hcf wfΣ. intros. eapply cumul_prop2; tea. now apply ws_cumul_pb_forget in X0. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_prop2 | 1,448 |
Γ A B u : Sort.is_sprop u -> isType Σ Γ A -> Σ ;;; Γ |- B : tSort u -> Σ ;;; Γ ⊢ A ≤ B -> Σ ;;; Γ |- A : tSort u. Proof using Hcf wfΣ. intros. eapply cumul_sprop1; tea. now apply ws_cumul_pb_forget in X1. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_sprop1 | 1,449 |
Γ A B u : Sort.is_sprop u -> isType Σ Γ B -> Σ ;;; Γ ⊢ A ≤ B -> Σ ;;; Γ |- A : tSort u -> Σ ;;; Γ |- B : tSort u. Proof using Hcf wfΣ. intros. eapply cumul_sprop2; tea. now apply ws_cumul_pb_forget in X0. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_sprop2 | 1,450 |
(Σ : global_env_ext) Γ A B u : wf_ext Σ -> Sort.is_propositional u -> isType Σ Γ B -> Σ ;;; Γ ⊢ A ≤ B -> Σ ;;; Γ |- A : tSort u -> Σ ;;; Γ |- B : tSort u. Proof. intros wf. intros pu isTy cum Ha. destruct u => //. eapply cumul_prop2; eauto. eapply cumul_sprop2; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_propositional | 1,451 |
forall (Σ : global_env_ext) (Γ : context) (L : list term) (u : sort) (x x0 : term), wf_ext Σ -> Sort.is_propositional u -> typing_spine Σ Γ x L x0 -> Σ;;; Γ |- x : tSort u -> ∑ u', Σ;;; Γ |- x0 : tSort u' × Sort.is_propositional u'. Proof. intros Σ Γ L u x x0 HΣ ? t1 c0. assert (X : wf Σ) by apply HΣ. revert u H c0. induction t1; intros. - destruct u => //. eapply cumul_prop2 in c0; eauto. eapply cumul_sprop2 in c0; eauto. - eapply cumul_propositional in c0; auto. 2-3: tea. eapply inversion_Prod in c0 as (? & ? & ? & ? & e0) ; auto. eapply ws_cumul_pb_Sort_inv in e0. eapply IHt1. 2: eapply @substitution0 with (T := tSort _); tea. unfold compare_sort, leq_sort in *. destruct u, x0, x => //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | sort_typing_spine | 1,452 |
(Σ : global_env_ext) Γ t T1 T2 : wf_ext Σ -> wf_local Σ Γ -> Σ ;;; Γ |- t : T1 -> isArity T1 -> Σ ;;; Γ |- t : T2 -> Is_conv_to_Arity Σ Γ T2. Proof. intros wfΣ wfΓ. intros. destruct (common_typing _ _ X X0) as (? & e & ? & ?). eapply invert_cumul_arity_l_gen; tea. eapply invert_cumul_arity_r_gen. 2:exact e. exists T1. split; auto. sq. eapply PCUICValidity.validity in X as (_ & s & Hs & _). eapply wt_closed_red_refl; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | arity_type_inv | 1,453 |
(Σ : global_env_ext) Γ A B u : wf_ext Σ -> isType Σ Γ A -> Sort.is_propositional u -> Σ ;;; Γ |- B : tSort u -> Σ ;;; Γ ⊢ A ≤ B -> Σ ;;; Γ |- A : tSort u. Proof. intros. destruct u => //. eapply cumul_prop1 in X2; eauto. eapply cumul_sprop1 in X2; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_prop1' | 1,454 |
(Σ : global_env_ext) Γ A B u : wf_ext Σ -> isType Σ Γ A -> Sort.is_propositional u -> Σ ;;; Γ |- B : tSort u -> Σ ;;; Γ ⊢ B ≤ A -> Σ ;;; Γ |- A : tSort u. Proof. intros. destruct u => //. eapply cumul_prop2 in X2; eauto. eapply cumul_sprop2 in X2; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | cumul_prop2' | 1,455 |
{Σ Γ v v' u u'} : wf_ext Σ -> PCUICEquality.leq_term Σ (global_ext_constraints Σ) v v' -> Σ;;; Γ |- v : tSort u -> Σ;;; Γ |- v' : tSort u' -> Sort.is_propositional u -> leq_sort (global_ext_constraints Σ) u' u. Proof. intros wf leq Hv Hv' isp. eapply leq_term_prop_sorted_l; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | leq_term_propositional_sorted_l | 1,456 |
{Σ Γ v v' u u'} : wf_ext Σ -> PCUICEquality.leq_term Σ (global_ext_constraints Σ) v v' -> Σ;;; Γ |- v : tSort u -> Σ;;; Γ |- v' : tSort u' -> Sort.is_propositional u' -> leq_sort (global_ext_constraints Σ) u u'. Proof. intros wfΣ leq hv hv' isp. eapply leq_term_prop_sorted_r; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | leq_term_propopositional_sorted_r | 1,457 |
(Σ : global_env_ext) Γ t L T : wf_ext Σ -> wf_local Σ Γ -> Σ ;;; Γ |- mkApps t L : T -> isErasable Σ Γ t -> ∥isErasable Σ Γ (mkApps t L)∥. Proof. intros wfΣ wfΓ ? ?. assert (HW : isType Σ Γ T). eapply PCUICValidity.validity; eauto. eapply PCUICValidity.inversion_mkApps in X as (? & ? & ?); auto. destruct X0 as (? & ? & [ | [u]]). - eapply common_typing in t2 as (? & e & e0 & ?). 2:eauto. 2:exact t0. eapply invert_cumul_arity_r in e0; eauto. destruct e0 as (? & ? & ?). destruct H as []. eapply ws_cumul_pb_red_l_inv in e. 2:exact X. eapply type_reduction_closed in t2; tea. eapply typing_spine_strengthen in t1. 3:tea. unshelve epose proof (isArity_typing_spine wfΓ t1). 2:{ eapply PCUICValidity.validity in t2; tea; pcuic. } forward H. eapply arity_type_inv; tea. destruct H as [T' [[]]]. sq. exists T'. split. eapply type_mkApps; tea. eapply typing_spine_weaken_concl; tea. now eapply red_conv. eapply isType_red; tea; pcuic. exact X0. now left. - destruct p. eapply PCUICPrincipality.common_typing in t2 as (? & e & e0 & ?). 2:eauto. 2:exact t0. eapply cumul_prop1' in e0; eauto. eapply cumul_propositional in e; eauto. econstructor. exists T. split. eapply type_mkApps. 2:eassumption. eassumption. right. eapply sort_typing_spine in t1; eauto. now eapply PCUICValidity.validity in t0. now apply PCUICValidity.validity in t2. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_type_app | 1,458 |
{cf : checker_flags} (ϕ : ConstraintSet.t) (u1 u2 : sort) : leq_sort ϕ u1 u2 -> Sort.is_propositional u2 -> Sort.is_propositional u1. Proof. destruct u1, u2 => //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | leq_sort_propositional_r | 1,459 |
{cf : checker_flags} (ϕ : ConstraintSet.t) (u1 u2 : sort) : prop_sub_type = false -> leq_sort ϕ u1 u2 -> Sort.is_propositional u1 -> Sort.is_propositional u2. Proof. destruct u1, u2 => //= -> //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | leq_sort_propositional_l | 1,460 |
x2 x3 : Sort.is_propositional (Sort.sort_of_product x2 x3) -> Sort.is_propositional x3. Proof. destruct x2, x3 => //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | is_propositional_sort_prod | 1,461 |
(Σ : global_env_ext) Γ na T1 t : wf_ext Σ -> wf_local Σ Γ -> isErasable Σ Γ (tLambda na T1 t) -> ∥isErasable Σ (vass na T1 :: Γ) t∥. Proof. intros ? ? (T & ? & ?). eapply inversion_Lambda in t0 as (? & h1 & ? & e); auto. destruct s as [ | (u & ? & ?)]. - eapply invert_cumul_arity_r in e; eauto. destruct e as (? & [] & ?). eapply invert_red_prod in X1 as (? & ? & []); eauto; subst. cbn in H. econstructor. exists x2. econstructor. eapply type_reduction_closed; eauto. econstructor; eauto. - sq. eapply cumul_prop1' in e; eauto. eapply inversion_Prod in e as (? & ? & ? & ? & e) ; auto. eapply ws_cumul_pb_Sort_inv in e. eapply leq_sort_propositional_r in e as H0; cbn; eauto. eexists. split. eassumption. right. eexists. split. eassumption. eapply is_propositional_sort_prod in H0; eauto. eapply type_Lambda in h1; eauto. now apply PCUICValidity.validity in h1. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_type_lambda | 1,462 |
(Σ : global_env_ext) Γ t v: wf Σ -> red Σ Γ t v -> isErasable Σ Γ t -> isErasable Σ Γ v. Proof. intros ? ? (T & ? & ?). exists T. split. - eapply subject_reduction; eauto. - eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_type_red | 1,463 |
(Σ : global_env_ext) t v: wf Σ -> eval Σ t v -> isErasable Σ [] t -> isErasable Σ [] v. Proof. intros; eapply Is_type_red. eauto. red in X1. destruct X1 as [T [HT _]]. eapply wcbveval_red; eauto. assumption. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_type_eval | 1,464 |
(Σ : global_env_ext) t v: wf_ext Σ -> welltyped Σ [] t -> PCUICWcbvEval.eval Σ t v -> isErasable Σ [] v -> ∥ isErasable Σ [] t ∥. Proof. intros wfΣ [T HT] ev [vt [Ht Hp]]. eapply wcbveval_red in ev; eauto. pose proof (subject_reduction _ _ _ _ _ wfΣ.1 HT ev). pose proof (common_typing _ wfΣ Ht X) as [P [Pvt [Pt vP]]]. destruct Hp. eapply arity_type_inv in X. 5:eauto. all:eauto. red in X. destruct X as [T' [[red] isA]]. eapply type_reduction_closed in HT; eauto. sq. exists T'; intuition auto. sq. exists T. intuition auto. right. destruct s as [u [vtu isp]]. exists u; intuition auto. eapply cumul_propositional; eauto. now eapply PCUICValidity.validity in HT. eapply cumul_prop1'; eauto. now eapply PCUICValidity.validity in vP. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_type_eval_inv | 1,465 |
{Σ} {wfΣ : wf Σ} {Γ T} : isType Σ Γ T -> Σ ;;; Γ ⊢ T ⇝ T. Proof. intros (_ & s & hs & _); eapply wt_closed_red_refl; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isType_closed_red_refl | 1,466 |
{Σ} {wfΣ : wf Σ} {Γ x} : ~ Is_conv_to_Arity Σ Γ x -> isWfArity Σ Γ x -> False. Proof. intros nis [isTy [ctx [s da]]]. apply nis. red. exists (it_mkProd_or_LetIn ctx (tSort s)). split. sq. apply destArity_spec_Some in da. simpl in da. subst x. eapply isType_closed_red_refl; pcuic. now eapply it_mkProd_isArity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | nIs_conv_to_Arity_isWfArity_elim | 1,467 |
(Σ : global_env_ext) Γ T := (Is_conv_to_Arity Σ Γ T + (∑ u : sort, Σ;;; Γ |- T : tSort u × Sort.is_propositional u))%type. | Definition | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isErasable_Type | 1,468 |
{Σ} {wfΣ : wf_ext Σ} {Γ t T} : isErasable Σ Γ t -> Σ ;;; Γ |- t : T -> isErasable_Type Σ Γ T. Proof. intros [T' [Ht Ha]]. intros HT. destruct (PCUICPrincipality.common_typing _ wfΣ Ht HT) as [P [le [le' tC]]]. sq. destruct Ha. left. eapply arity_type_inv. 3:exact Ht. all:eauto using typing_wf_local. destruct s as [u [Hu isp]]. right. exists u; split; auto. eapply cumul_propositional; eauto. eapply PCUICValidity.validity; eauto. eapply cumul_prop1'; eauto. eapply PCUICValidity.validity; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isErasable_any_type | 1,469 |
Σ Γ t : wf_ext Σ -> Is_proof Σ Γ t -> forall t' ty, Σ ;;; Γ |- t : ty -> Σ ;;; Γ |- t' : ty -> Is_proof Σ Γ t'. Proof. intros wfΣ [ty [u [Hty isp]]]. intros t' ty' Hty'. epose proof (PCUICPrincipality.common_typing _ wfΣ Hty Hty') as [C [Cty [Cty' Ht'']]]. intros Ht'. exists ty', u; intuition auto. eapply PCUICValidity.validity in Hty; eauto. eapply PCUICValidity.validity in Hty'; eauto. eapply PCUICValidity.validity in Ht''; eauto. eapply cumul_prop1' in Cty; eauto. eapply cumul_propositional in Cty'; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_proof_ty | 1,470 |
{Σ Γ T s s'} : wf_ext Σ -> prop_sub_type = false -> Σ ;;; Γ |- T ~~ tSort s -> Σ ;;; Γ |- T ~~ tSort s' -> PCUICCumulProp.eq_univ_prop s s'. Proof. intros. eapply PCUICCumulProp.cumul_prop_sort. etransitivity; tea. now symmetry. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | is_propositional_bottom' | 1,471 |
{Σ Γ T s s'} : wf_ext Σ -> prop_sub_type = false -> Σ ;;; Γ ⊢ T ≤ tSort s -> Σ ;;; Γ ⊢ T ≤ tSort s' -> PCUICCumulProp.eq_univ_prop s s'. Proof. move => wf pst /cumul_pb_cumul_prop h /cumul_pb_cumul_prop h'. now eapply is_propositional_bottom'. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | is_propositional_bottom | 1,472 |
{Σ s u u'} : leq_sort Σ s u -> leq_sort Σ s u' -> PCUICCumulProp.eq_univ_prop u u'. Proof. destruct s, u, u' => //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | is_propositional_lower | 1,473 |
{Σ Γ Δ s args args' u u'} : wf_ext Σ -> prop_sub_type = false -> let T := it_mkProd_or_LetIn Δ (tSort s) in typing_spine Σ Γ T args (tSort u) -> typing_spine Σ Γ T args' (tSort u') -> PCUICCumulProp.eq_univ_prop u u'. Proof. intros wf ips T. move/typing_spine_it_mkProd_or_LetIn_full_inv => su. move/typing_spine_it_mkProd_or_LetIn_full_inv => su'. eapply is_propositional_lower; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | typing_spine_inj | 1,474 |
Σ Γ t : wf_ext Σ -> Is_proof Σ Γ t -> forall t' ind u args args', Σ ;;; Γ |- t : mkApps (tInd ind u) args -> Σ ;;; Γ |- t' : mkApps (tInd ind u) args' -> Is_proof Σ Γ t'. Proof. intros wfΣ [ty [u [Hty isp]]]. intros t' ind u' args args' Hty' Hty''. epose proof (PCUICPrincipality.common_typing _ wfΣ Hty Hty') as [C [Cty [Cty' Ht'']]]. destruct isp. assert (Σ ;;; Γ |- C : tSort u). eapply cumul_prop1'; tea => //. now eapply validity. assert (Σ ;;; Γ |- mkApps (tInd ind u') args : tSort u). eapply cumul_prop2'; tea => //. now eapply validity. eapply inversion_mkApps in X0 as x1. destruct x1 as [? []]. eapply inversion_Ind in t1 as [mdecl [idecl [wf [decli ?]]]]; eauto. destruct (validity Hty'') as (_ & u'' & tyargs' & _). eapply inversion_mkApps in X0 as x1. destruct x1 as [? []]. eapply invert_type_mkApps_ind in X0 as [sp cum]; eauto. eapply invert_type_mkApps_ind in tyargs' as f; tea. destruct f as [sp' cum']; eauto. do 2 eexists. split => //. tea. instantiate (1 := u''). split => //. rewrite (declared_inductive_type decli) in sp, sp'. rewrite subst_instance_it_mkProd_or_LetIn /= in sp, sp'. eapply typing_spine_inj in sp. 4:exact sp'. all:eauto. destruct u, u'' => //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_proof_ind | 1,475 |
{Σ : global_env_ext} {Γ ip p discr discr' brs T} {wfΣ : wf_ext Σ} : PCUICReduction.red Σ Γ (tCase ip p discr brs) (tCase ip p discr' brs) -> Σ ;;; Γ |- tCase ip p discr brs : T -> Is_proof Σ Γ discr -> Is_proof Σ Γ discr'. Proof. intros hr hc. eapply subject_reduction in hr; tea; eauto. eapply inversion_Case in hc as [mdecl [idecl [isdecl [indices ?]]]]; eauto. eapply inversion_Case in hr as [mdecl' [idecl' [isdecl' [indices' ?]]]]; eauto. pose proof (wfΣ' := wfΣ.1). unshelve eapply declared_inductive_to_gen in isdecl, isdecl'; eauto. destruct (declared_inductive_inj isdecl isdecl'). subst mdecl' idecl'. intros hp. epose proof (Is_proof_ind _ _ _ wfΣ hp). destruct p0 as [[] ?]. destruct p1 as [[] ?]. exact (X _ _ _ _ _ scrut_ty scrut_ty0). Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | red_case_isproof | 1,476 |
{Σ Γ t args ty} {wfΣ : wf_ext Σ} : Is_proof Σ Γ t -> Σ ;;; Γ |- mkApps t args : ty -> Is_proof Σ Γ (mkApps t args). Proof. intros [ty' [u [Hty [isp pu]]]] Htargs. eapply PCUICValidity.inversion_mkApps in Htargs as [A [Ht sp]]. pose proof (PCUICValidity.validity Hty). pose proof (PCUICValidity.validity Ht). epose proof (PCUICPrincipality.common_typing _ wfΣ Hty Ht) as [C [Cty [Cty' Ht'']]]. eapply PCUICSpine.typing_spine_strengthen in sp. 3:tea. edestruct (sort_typing_spine _ _ _ u _ _ _ pu sp) as [u' [Hty' isp']]. eapply cumul_prop1'. 4:tea. all:eauto. eapply validity; eauto. exists ty, u'; split; auto. eapply PCUICSpine.type_mkApps; tea; eauto. now eapply validity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Is_proof_app | 1,477 |
{Σ : global_env_ext} {Γ ind n u args} : wf_ext Σ -> isErasable Σ Γ (mkApps (tConstruct ind n u) args) -> isPropositional Σ ind. Proof. intros wfΣ ise. eapply tConstruct_no_Type in ise; eauto. destruct ise as [T [s [HT [Ts isp]]]]. unfold isPropositional. eapply PCUICValidity.inversion_mkApps in HT as (? & ? & ?); auto. eapply inversion_Construct in t as (? & ? & ? & ? & ? & ? & ?); auto. pose proof (wfΣ' := wfΣ.1). unshelve epose proof (d_ := declared_constructor_to_gen d); eauto. unfold lookup_inductive. rewrite (declared_inductive_lookup_gen d_.p1). destruct (on_declared_constructor d). destruct p as [onind oib]. rewrite oib.(ind_arity_eq). rewrite /isPropositionalArity !destArity_it_mkProd_or_LetIn /=. eapply PCUICSpine.typing_spine_strengthen in t0; eauto. unfold type_of_constructor in t0. destruct s0 as [indctors [nthcs onc]]. rewrite onc.(cstr_eq) in t0. rewrite !subst_instance_it_mkProd_or_LetIn !PCUICLiftSubst.subst_it_mkProd_or_LetIn in t0. len in t0. rewrite subst_cstr_concl_head in t0. destruct d as [decli declc]. destruct decli as [declm decli]. now eapply nth_error_Some_length. rewrite -it_mkProd_or_LetIn_app in t0. eapply PCUICElimination.typing_spine_proofs in Ts; eauto. destruct Ts as [_ Hs]. specialize (Hs _ _ d c) as [Hs _]. specialize (Hs isp). subst s. move: isp. now destruct (ind_sort x1). eapply validity. econstructor; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isErasable_Propositional | 1,478 |
{Σ : global_env_ext} {Γ ind n u} : wf_ext Σ -> welltyped Σ Γ (tConstruct ind n u) -> (isErasable Σ Γ (tConstruct ind n u) -> False) -> ~~ isPropositional Σ ind. Proof. intros wfΣ wt ise. destruct wt as [T HT]. epose proof HT as HT'. eapply inversion_Construct in HT' as (? & ? & ? & ? & ? & ? & e); auto. pose proof (declared_constructor_valid_ty _ _ _ _ _ _ _ _ wfΣ a d c). pose proof d as [decli ?]. destruct (on_declared_constructor d). destruct p as [onind oib]. unfold isPropositional, lookup_inductive. pose proof (wfΣ' := wfΣ.1). unshelve epose proof (decli_ := declared_inductive_to_gen decli); eauto. rewrite (declared_inductive_lookup_gen decli_). rewrite oib.(ind_arity_eq). rewrite /isPropositionalArity !destArity_it_mkProd_or_LetIn /=. destruct (Sort.is_propositional (ind_sort x0)) eqn:isp; auto. exfalso; eapply ise. red. eexists; intuition eauto. right. unfold type_of_constructor in e, X. destruct s as [indctors [nthcs onc]]. rewrite onc.(cstr_eq) in e, X. rewrite !subst_instance_it_mkProd_or_LetIn !PCUICLiftSubst.subst_it_mkProd_or_LetIn in e, X. len in e; len in X. rewrite subst_cstr_concl_head in e, X. destruct decli. eapply nth_error_Some_length in H1; eauto. rewrite -it_mkProd_or_LetIn_app in e, X. exists (subst_instance_sort u (ind_sort x0)). rewrite is_propositional_subst_instance => //. split; auto. eapply cumul_propositional; eauto. rewrite is_propositional_subst_instance => //. eapply PCUICValidity.validity; eauto. destruct X as (_ & cty & ty & _). eapply type_Cumul_alt; eauto. eapply isType_Sort. 2:eauto. destruct (ind_sort x0) => //. eapply PCUICSpine.inversion_it_mkProd_or_LetIn in ty; eauto. epose proof (typing_spine_proofs _ _ [] _ _ _ [] _ _ wfΣ ty). forward H0 by constructor. eapply has_sort_isType; eauto. simpl. now eapply has_sort_isType. eapply PCUICConversion.ws_cumul_pb_eq_le_gen, PCUICSR.wt_cumul_pb_refl; eauto. destruct H0 as [_ sorts]. specialize (sorts _ _ decli c) as [sorts sorts']. forward sorts' by constructor. do 2 constructor. rewrite is_propositional_subst_instance in sorts, sorts' |- *. specialize (sorts' isp). rewrite -sorts'. reflexivity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | nisErasable_Propositional | 1,479 |
Σ {wfΣ: wf Σ} (Σ' : E.global_context) ind mdecl idecl mdecl' idecl' : PCUICAst.declared_inductive Σ ind mdecl idecl -> EGlobalEnv.declared_inductive Σ' ind mdecl' idecl' -> erases_mutual_inductive_body mdecl mdecl' -> erases_one_inductive_body idecl idecl' -> EGlobalEnv.inductive_isprop_and_pars Σ' ind = Some (isPropositional Σ ind, mdecl.(ind_npars)). Proof. intros decli decli' [_ indp] []. unfold isPropositional, EGlobalEnv.inductive_isprop_and_pars. unfold lookup_inductive. unshelve epose proof (decli_ := declared_inductive_to_gen decli); eauto. rewrite (declared_inductive_lookup_gen decli_). rewrite (EGlobalEnv.declared_inductive_lookup decli') /= /isPropositionalArity. destruct H0 as [_ [_ [_ isP]]]. unfold isPropositionalArity in isP. destruct destArity as [[ctx s]|] eqn:da => //. rewrite isP; congruence. congruence. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isPropositional_propositional | 1,480 |
Σ (Σ' : E.global_context) ind c mdecl idecl cdecl mdecl' idecl' : wf Σ -> PCUICAst.declared_constructor Σ (ind, c) mdecl idecl cdecl -> EGlobalEnv.declared_inductive Σ' ind mdecl' idecl' -> erases_mutual_inductive_body mdecl mdecl' -> erases_one_inductive_body idecl idecl' -> EGlobalEnv.constructor_isprop_pars_decl Σ' ind c = Some (isPropositional Σ ind, mdecl.(ind_npars), EAst.mkConstructor cdecl.(cstr_name) (context_assumptions cdecl.(cstr_args))). Proof. intros wfΣ declc decli' em ei. pose proof declc as [decli'' _]. eapply isPropositional_propositional in decli''; tea. move: decli''. rewrite /EGlobalEnv.inductive_isprop_and_pars. unfold EGlobalEnv.constructor_isprop_pars_decl. unfold EGlobalEnv.lookup_constructor. rewrite (EGlobalEnv.declared_inductive_lookup decli') /=. intros [= <- <-]. destruct ei. clear H0. eapply Forall2_nth_error_Some in H as [cdecl' []]; tea. 2:apply declc. rewrite H //. f_equal. f_equal. destruct cdecl'. cbn in *. destruct H0. subst. f_equal. destruct (on_declared_constructor declc) as [[] [? []]]. now eapply cstr_args_length in o1. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isPropositional_propositional_cstr | 1,481 |
{cf : checker_flags} {Σ : global_env_ext} ci p discr brs res T : wf Σ -> Σ ;;; [] |- tCase ci p discr brs : T -> eval Σ (tCase ci p discr brs) res -> ∑ c u args, PCUICReduction.red Σ [] (tCase ci p discr brs) (tCase ci p ((mkApps (tConstruct ci.(ci_ind) c u) args)) brs). Proof. intros wf wt H. depind H; try now (cbn in *; congruence). - eapply inversion_Case in wt as (? & ? & ? & ? & cinv & ?); eauto. eexists _, _, _. eapply PCUICReduction.red_case_c. eapply wcbveval_red. 2: eauto. eapply cinv. - eapply inversion_Case in wt as wt'; eauto. destruct wt' as (? & ? & ? & ? & cinv & ?). assert (Hred1 : PCUICReduction.red Σ [] (tCase ip p discr brs) (tCase ip p (mkApps fn args) brs)). { etransitivity. { eapply PCUICReduction.red_case_c. eapply wcbveval_red. 2: eauto. eapply cinv. } econstructor. econstructor. rewrite closed_unfold_cofix_cunfold_eq. eauto. enough (closed (mkApps (tCoFix mfix idx) args)) as Hcl by (rewrite closedn_mkApps in Hcl; solve_all). eapply eval_closed. eauto. 2: eauto. eapply @PCUICClosedTyp.subject_closed with (Γ := []); eauto. eapply cinv. eauto. } edestruct IHeval2 as (c & u & args0 & IH); eauto using subject_reduction. exists c, u, args0. etransitivity; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | eval_tCase | 1,482 |
v ci p brs T (Σ : global_env_ext) : wf_ext Σ -> forall (mdecl : mutual_inductive_body) (idecl : one_inductive_body) mfix idx, declared_inductive Σ.1 ci.(ci_ind) mdecl idecl -> forall (args : list term), Subsingleton Σ ci.(ci_ind) -> Σ ;;; [] |- tCase ci p (mkApps (tCoFix mfix idx) args) brs : T -> Σ ⊢p tCase ci p (mkApps (tCoFix mfix idx) args) brs ⇓ v -> Is_proof Σ [] (mkApps (tCoFix mfix idx) args) -> #|ind_ctors idecl| <= 1. Proof. intros. destruct Σ as [Σ1 Σ2]. cbn in *. eapply eval_tCase in X0 as X2'; eauto. destruct X2' as (? & ? & ? & ?). eapply subject_reduction in X0 as X2'; eauto. eapply inversion_Case in X2' as (? & ? & ? & ? & [] & ?); eauto. eapply inversion_Case in X0 as (? & ? & ? & ? & [] & ?); eauto. pose (X' := X.1). unshelve eapply declared_inductive_to_gen in x8, x4, H; eauto. destruct (declared_inductive_inj x8 x4); subst. destruct (declared_inductive_inj x8 H); subst. eapply H0; eauto. apply declared_inductive_from_gen; eauto. reflexivity. eapply Is_proof_ind; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | Subsingleton_cofix | 1,483 |
{Σ : global_env_ext} {Γ mfix idx} {wfΣ : wf Σ} decl : isErasable Σ Γ (tCoFix mfix idx) -> nth_error mfix idx = Some decl -> isErasable Σ Γ (subst0 (cofix_subst mfix) (dbody decl)). Proof. intros [Tty []] hred. exists Tty. split => //. eapply type_tCoFix_inv in t as t''; eauto. destruct t'' as [decl' [[[] h'] h'']]. rewrite e in hred. noconf hred. eapply type_ws_cumul_pb; tea. now eapply validity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isErasable_unfold_cofix | 1,484 |
{Σ : global_env_ext} {Γ T U} {wfΣ : wf Σ} : isErasable Σ Γ T -> PCUICReduction.red Σ Γ T U -> isErasable Σ Γ U. Proof. intros [Tty []] hred. exists Tty. split => //. eapply subject_reduction; tea. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | isErasable_red | 1,485 |
Σ Γ f A u : wf_ext Σ -> wf_local Σ Γ -> ∥Σ;;; Γ |- f : A∥ -> (forall B, ∥Σ;;; Γ ⊢ A ⇝ B∥ -> A = B) -> (forall B, ∥Σ ;;; Γ |- f : B∥ -> ∥Σ ;;; Γ ⊢ A ≤ B∥) -> ~ ∥ isArity A ∥ -> ∥ Σ;;; Γ |- A : tSort u ∥ -> ~ Sort.is_propositional u -> ~ ∥ Extract.isErasable Σ Γ f ∥. Proof. intros wfΣ Hlocal Hf Hnf Hprinc Harity Hfu Hu [[T [HT []]]]; sq. - eapply Harity; sq. eapply EArities.arity_type_inv in i as [T' [? ?]]; eauto. eapply Hnf in H. subst; eauto. - destruct s as [s [? ?]]. eapply Hu. specialize (Hprinc _ (sq HT)). pose proof (Hs := i). sq. eapply PCUICElimination.unique_sorting_equality_propositional in Hprinc; eauto. rewrite Hprinc; eauto. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | not_isErasable | 1,486 |
p prim_ty : ∑ u args, prim_type p prim_ty = mkApps (tConst prim_ty u) args. Proof. destruct p as [? []]; simp prim_type. - eexists [], []. reflexivity. - eexists [], []; reflexivity. - eexists [], []; reflexivity. - eexists [_], [_]; reflexivity. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | prim_type_inv | 1,487 |
t decl : primitive_invariants t decl -> cst_body decl = None. Proof. destruct t; cbn => //. 1-3:now intros [? []]. now intros []. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | primitive_invariants_axiom | 1,488 |
Σ p : wf_ext Σ -> isErasable Σ [] (tPrim p) -> False. Proof. intros wfΣ [T [Ht h]]. eapply inversion_Prim in Ht as [prim_ty [cdecl []]]; eauto. pose proof (type_Prim _ _ _ _ _ a e d p0 p1). eapply validity in X. destruct h. - eapply invert_cumul_arity_r in w; tea. destruct w as [ar [[H] r]]. destruct (prim_type_inv p prim_ty) as [u [args eq]]. rewrite eq in H. eapply invert_red_axiom_app in H as [args' []]; tea. 2:now eapply primitive_invariants_axiom. subst ar. now eapply isArity_mkApps in r as []. - destruct s as [s [hs isp]]. eapply cumul_prop1' in hs; tea; eauto. depelim p1; simp prim_type in hs. * destruct p0 as [hd hb hu]. eapply inversion_Const in hs as [decl' [wf [decl'' [cu hs']]]]; eauto. unshelve eapply declared_constant_to_gen in d, decl''. 3,6:eapply wfΣ. eapply declared_constant_inj in d; tea. subst decl'. rewrite hd in hs'. cbn in hs'. eapply ws_cumul_pb_Sort_inv in hs'. red in hs'. destruct s => //. * destruct p0 as [hd hb hu]. eapply inversion_Const in hs as [decl' [wf [decl'' [cu hs']]]]; eauto. unshelve eapply declared_constant_to_gen in d, decl''. 3,6:eapply wfΣ. eapply declared_constant_inj in d; tea. subst decl'. rewrite hd in hs'. cbn in hs'. eapply ws_cumul_pb_Sort_inv in hs'. red in hs'. destruct s => //. * destruct p0 as [hd hb hu]. eapply inversion_Const in hs as [decl' [wf [decl'' [cu hs']]]]; eauto. unshelve eapply declared_constant_to_gen in d, decl''. 3,6:eapply wfΣ. eapply declared_constant_inj in d; tea. subst decl'. rewrite hd in hs'. cbn in hs'. eapply ws_cumul_pb_Sort_inv in hs'. red in hs'. destruct s => //. * destruct p0 as [hd hb hu]. eapply inversion_App in hs as [na [A [B [hp [harg hres]]]]]; eauto. eapply inversion_Const in hp as [decl' [wf [decl'' [cu hs']]]]; eauto. unshelve eapply declared_constant_to_gen in d, decl''. 3,6:eapply wfΣ. eapply declared_constant_inj in d; tea. subst decl'. rewrite hd in hs'. cbn in hs'. eapply ws_cumul_pb_Prod_Prod_inv in hs' as []. eapply substitution_ws_cumul_pb_vass in w1; tea. cbn in w1. pose proof (ws_cumul_pb_trans _ _ _ w1 hres) as X0. eapply ws_cumul_pb_Sort_inv in X0. destruct s => //. Qed. | Lemma | erasure | From Coq Require Import ssreflect ssrbool. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import config. From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils From MetaCoq.Erasure Require Import Extract. Require Import Program. From Equations Require Import Equations. Import PCUICGlobalEnv PCUICUnivSubst PCUICValidity PCUICCumulProp. | erasure\theories\EArities.v | nisErasable_tPrim | 1,489 |
(term : Set) := { dname : name; dbody : term; rarg : nat }. | Record | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | def | 1,490 |
{term : Set} (f : term -> term) (d : def term) := {| dname := d.(dname); dbody := f d.(dbody); rarg := d.(rarg) |}. | Definition | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | map_def | 1,491 |
{term : Set} (f : term -> bool) (d : def term) := f d.(dbody). | Definition | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | test_def | 1,492 |
(term : Set) := list (def term). | Definition | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | mfixpoint | 1,493 |
Set := | tBox | tRel (n : nat) | tVar (i : ident) (* For free variables (e.g. | Inductive | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | term | 1,494 |
t us := match us with | nil => t | a :: args => mkApps (tApp t a) args end. | Fixpoint | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | mkApps | 1,495 |
t u := Eval cbn in mkApps t [u]. | Definition | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | mkApp | 1,496 |
t := match t with | tApp _ _ => true | _ => false end. | Definition | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | isApp | 1,497 |
t := match t with | tLambda _ _ => true | _ => false end. | Definition | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | isLambda | 1,498 |
{ }. | Record | erasure | From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Import BasicAst Universes. From MetaCoq.Erasure Require Import EPrimitive. | erasure\theories\EAst.v | parameter_entry | 1,499 |
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