Mcltt.Core.Soundness.LogicalRelation.Definitions
From Coq Require Import Relation_Definitions RelationClasses.
From Equations Require Import Equations.
From Mcltt Require Import Base LibTactics.
From Mcltt.Core Require Import System.Definitions.
From Mcltt.Core Require Export Domain PER.Definitions Soundness.Weakening.Definition.
Import Domain_Notations.
Global Open Scope predicate_scope.
Generalizable All Variables.
Notation "'glu_typ_pred_args'" := (Tcons ctx (Tcons typ Tnil)).
Notation "'glu_typ_pred'" := (predicate glu_typ_pred_args).
Notation "'glu_typ_pred_equivalence'" := (@predicate_equivalence glu_typ_pred_args) (only parsing).
From Equations Require Import Equations.
From Mcltt Require Import Base LibTactics.
From Mcltt.Core Require Import System.Definitions.
From Mcltt.Core Require Export Domain PER.Definitions Soundness.Weakening.Definition.
Import Domain_Notations.
Global Open Scope predicate_scope.
Generalizable All Variables.
Notation "'glu_typ_pred_args'" := (Tcons ctx (Tcons typ Tnil)).
Notation "'glu_typ_pred'" := (predicate glu_typ_pred_args).
Notation "'glu_typ_pred_equivalence'" := (@predicate_equivalence glu_typ_pred_args) (only parsing).
This type annotation is to distinguish this notation from others
Notation "Γ ⊢ A ® R" := ((R Γ A : (Prop : Type)) : (Prop : (Type : Type))) (in custom judg at level 80, Γ custom exp, A custom exp, R constr).
Notation "'glu_exp_pred_args'" := (Tcons ctx (Tcons typ (Tcons exp (Tcons domain Tnil)))).
Notation "'glu_exp_pred'" := (predicate glu_exp_pred_args).
Notation "'glu_exp_pred_equivalence'" := (@predicate_equivalence glu_exp_pred_args) (only parsing).
Notation "Γ ⊢ M : A ® m ∈ R" := (R Γ A M m : (Prop : (Type : Type))) (in custom judg at level 80, Γ custom exp, M custom exp, A custom exp, m custom domain, R constr).
Notation "'glu_sub_pred_args'" := (Tcons ctx (Tcons sub (Tcons env Tnil))).
Notation "'glu_sub_pred'" := (predicate glu_sub_pred_args).
Notation "'glu_sub_pred_equivalence'" := (@predicate_equivalence glu_sub_pred_args) (only parsing).
Notation "Γ ⊢s σ ® ρ ∈ R" := ((R Γ σ ρ : Prop) : (Prop : (Type : Type))) (in custom judg at level 80, Γ custom exp, σ custom exp, ρ custom domain, R constr).
Notation "'DG' a ∈ R ↘ P ↘ El" := (R P El a : ((Prop : Type) : (Type : Type))) (in custom judg at level 90, a custom domain, R constr, P constr, El constr).
Notation "'EG' A ∈ R ↘ Sb " := (R Sb A : ((Prop : (Type : Type)) : (Type : Type))) (in custom judg at level 90, A custom exp, R constr, Sb constr).
Inductive glu_nat : ctx -> exp -> domain -> Prop :=
| glu_nat_zero :
`{ {{ Γ ⊢ M ≈ zero : ℕ }} ->
glu_nat Γ M d{{{ zero }}} }
| glu_nat_succ :
`{ {{ Γ ⊢ M ≈ succ M' : ℕ }} ->
glu_nat Γ M' m' ->
glu_nat Γ M d{{{ succ m' }}} }
| glu_nat_neut :
`{ per_bot m m ->
(forall {Δ σ M'}, {{ Δ ⊢w σ : Γ }} -> {{ Rne m in length Δ ↘ M' }} -> {{ Δ ⊢ M[σ] ≈ M' : ℕ }}) ->
glu_nat Γ M d{{{ ⇑ ℕ m }}} }.
#[export]
Hint Constructors glu_nat : mcltt.
Definition nat_glu_typ_pred i : glu_typ_pred := fun Γ A => {{ Γ ⊢ A ≈ ℕ : Type@i }}.
Arguments nat_glu_typ_pred i Γ A/.
Definition nat_glu_exp_pred i : glu_exp_pred := fun Γ A M m => {{ Γ ⊢ A ® nat_glu_typ_pred i }} /\ glu_nat Γ M m.
Arguments nat_glu_exp_pred i Γ A M m/.
Definition neut_glu_typ_pred i a : glu_typ_pred :=
fun Γ A => {{ Γ ⊢ A : Type@i }} /\
(forall Δ σ A', {{ Δ ⊢w σ : Γ }} -> {{ Rne a in length Δ ↘ A' }} -> {{ Δ ⊢ A[σ] ≈ A' : Type@i }}).
Arguments neut_glu_typ_pred i a Γ A/.
Variant neut_glu_exp_pred i a : glu_exp_pred :=
| mk_neut_glu_exp_pred :
`{ {{ Γ ⊢ A ® neut_glu_typ_pred i a }} ->
{{ Γ ⊢ M : A }} ->
{{ Dom m ≈ m ∈ per_bot }} ->
(forall Δ σ M', {{ Δ ⊢w σ : Γ }} ->
{{ Rne m in length Δ ↘ M' }} ->
{{ Δ ⊢ M[σ] ≈ M' : A[σ] }}) ->
{{ Γ ⊢ M : A ® ⇑ b m ∈ neut_glu_exp_pred i a }} }.
Variant pi_glu_typ_pred i
(IR : relation domain)
(IP : glu_typ_pred)
(IEl : glu_exp_pred)
(OP : forall c (equiv_c : {{ Dom c ≈ c ∈ IR }}), glu_typ_pred) : glu_typ_pred :=
| mk_pi_glu_typ_pred :
`{ {{ Γ ⊢ A ≈ Π IT OT : Type@i }} ->
{{ Γ ⊢ IT : Type@i }} ->
{{ Γ , IT ⊢ OT : Type@i }} ->
(forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
(forall Δ σ M m,
{{ Δ ⊢w σ : Γ }} ->
{{ Δ ⊢ M : IT[σ] ® m ∈ IEl }} ->
forall (equiv_m : {{ Dom m ≈ m ∈ IR }}),
{{ Δ ⊢ OT[σ,,M] ® OP _ equiv_m }}) ->
{{ Γ ⊢ A ® pi_glu_typ_pred i IR IP IEl OP }} }.
Variant pi_glu_exp_pred i
(IR : relation domain)
(IP : glu_typ_pred)
(IEl : glu_exp_pred)
(elem_rel : relation domain)
(OEl : forall c (equiv_c : {{ Dom c ≈ c ∈ IR }}), glu_exp_pred): glu_exp_pred :=
| mk_pi_glu_exp_pred :
`{ {{ Γ ⊢ M : A }} ->
{{ Dom m ≈ m ∈ elem_rel }} ->
{{ Γ ⊢ A ≈ Π IT OT : Type@i }} ->
{{ Γ ⊢ IT : Type@i }} ->
{{ Γ , IT ⊢ OT : Type@i }} ->
(forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
(forall Δ σ N n,
{{ Δ ⊢w σ : Γ }} ->
{{ Δ ⊢ N : IT[σ] ® n ∈ IEl }} ->
forall (equiv_n : {{ Dom n ≈ n ∈ IR }}),
exists mn, {{ $| m & n |↘ mn }} /\ {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl _ equiv_n }}) ->
{{ Γ ⊢ M : A ® m ∈ pi_glu_exp_pred i IR IP IEl elem_rel OEl }} }.
#[export]
Hint Constructors neut_glu_exp_pred pi_glu_typ_pred pi_glu_exp_pred : mcltt.
Definition univ_glu_typ_pred j i : glu_typ_pred := fun Γ A => {{ Γ ⊢ A ≈ Type@j : Type@i }}.
Arguments univ_glu_typ_pred j i Γ A/.
Transparent univ_glu_typ_pred.
Section Gluing.
Variable
(i : nat)
(glu_univ_typ_rec : forall {j}, j < i -> domain -> glu_typ_pred).
Definition univ_glu_exp_pred' {j} (lt_j_i : j < i) : glu_exp_pred :=
fun Γ A M m =>
{{ Γ ⊢ M : A }} /\
{{ Γ ⊢ A ≈ Type@j : Type@i }} /\
{{ Γ ⊢ M ® glu_univ_typ_rec lt_j_i m }}.
#[global]
Arguments univ_glu_exp_pred' {j} lt_j_i Γ A M m/.
Inductive glu_univ_elem_core : glu_typ_pred -> glu_exp_pred -> domain -> Prop :=
| glu_univ_elem_core_univ :
`{ forall typ_rel
el_rel
(lt_j_i : j < i),
typ_rel <∙> univ_glu_typ_pred j i ->
el_rel <∙> univ_glu_exp_pred' lt_j_i ->
{{ DG 𝕌@j ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
| glu_univ_elem_core_nat :
`{ forall typ_rel el_rel,
typ_rel <∙> nat_glu_typ_pred i ->
el_rel <∙> nat_glu_exp_pred i ->
{{ DG ℕ ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
| glu_univ_elem_core_pi :
`{ forall (in_rel : relation domain)
IP IEl
(OP : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_typ_pred)
(OEl : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_exp_pred)
typ_rel el_rel
(elem_rel : relation domain),
{{ DG a ∈ glu_univ_elem_core ↘ IP ↘ IEl }} ->
{{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
(forall {c} (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) b,
{{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
{{ DG b ∈ glu_univ_elem_core ↘ OP _ equiv_c ↘ OEl _ equiv_c }}) ->
{{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
typ_rel <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
el_rel <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
{{ DG Π a ρ B ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
| glu_univ_elem_core_neut :
`{ forall typ_rel el_rel,
{{ Dom b ≈ b ∈ per_bot }} ->
typ_rel <∙> neut_glu_typ_pred i b ->
el_rel <∙> neut_glu_exp_pred i b ->
{{ DG ⇑ a b ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }.
End Gluing.
#[export]
Hint Constructors glu_univ_elem_core : mcltt.
Equations glu_univ_elem (i : nat) : glu_typ_pred -> glu_exp_pred -> domain -> Prop by wf i :=
| i => glu_univ_elem_core i (fun j lt_j_i a Γ A => exists P El, {{ DG a ∈ glu_univ_elem j ↘ P ↘ El }} /\ {{ Γ ⊢ A ® P }}).
Definition glu_univ_typ (i : nat) (a : domain) : glu_typ_pred :=
fun Γ A => exists P El, {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} /\ {{ Γ ⊢ A ® P }}.
Arguments glu_univ_typ i a Γ A/.
Definition univ_glu_exp_pred j i : glu_exp_pred :=
fun Γ A M m =>
{{ Γ ⊢ M : A }} /\ {{ Γ ⊢ A ≈ Type@j : Type@i }} /\
{{ Γ ⊢ M ® glu_univ_typ j m }}.
Arguments univ_glu_exp_pred j i Γ A M m/.
Section GluingInduction.
Hypothesis
(motive : nat -> glu_typ_pred -> glu_exp_pred -> domain -> Prop)
(case_univ :
forall i j
(P : glu_typ_pred) (El : glu_exp_pred) (lt_j_i : j < i),
(forall P' El' a, {{ DG a ∈ glu_univ_elem j ↘ P' ↘ El' }} -> motive j P' El' a) ->
P <∙> univ_glu_typ_pred j i ->
El <∙> univ_glu_exp_pred j i ->
motive i P El d{{{ 𝕌 @ j }}})
(case_nat :
forall i (P : glu_typ_pred) (El : glu_exp_pred),
P <∙> nat_glu_typ_pred i ->
El <∙> nat_glu_exp_pred i ->
motive i P El d{{{ ℕ }}})
(case_pi :
forall i a B (ρ : env) (in_rel : relation domain) (IP : glu_typ_pred)
(IEl : glu_exp_pred) (OP : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_typ_pred)
(OEl : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_exp_pred) (P : glu_typ_pred) (El : glu_exp_pred)
(elem_rel : relation domain),
{{ DG a ∈ glu_univ_elem i ↘ IP ↘ IEl }} ->
motive i IP IEl a ->
{{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
(forall (c : domain) (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) (b : domain),
{{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
{{ DG b ∈ glu_univ_elem i ↘ OP c equiv_c ↘ OEl c equiv_c }}) ->
(forall (c : domain) (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) (b : domain),
{{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
motive i (OP c equiv_c) (OEl c equiv_c) b) ->
{{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
P <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
El <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
motive i P El d{{{ Π a ρ B }}})
(case_neut :
forall i b a
(P : glu_typ_pred)
(El : glu_exp_pred),
{{ Dom b ≈ b ∈ per_bot }} ->
P <∙> neut_glu_typ_pred i b ->
El <∙> neut_glu_exp_pred i b ->
motive i P El d{{{ ⇑ a b }}})
.
#[local]
Ltac def_simp := simp glu_univ_elem in *; mauto 3.
#[derive(equations=no, eliminator=no), tactic="def_simp"]
Equations glu_univ_elem_ind i P El a
(H : glu_univ_elem i P El a) : motive i P El a by wf i :=
| i, P, El, a, H =>
glu_univ_elem_core_ind
i
(fun j lt_j_i a Γ A => glu_univ_typ j a Γ A)
(motive i)
(fun j P' El' lt_j_i HP' HEl' =>
case_univ i j P' El' lt_j_i
(fun P'' El'' A H => glu_univ_elem_ind j P'' El'' A H)
HP'
HEl')
(case_nat i)
_ (* (case_pi i) *)
(case_neut i)
P El a
_.
Next Obligation.
eapply (case_pi i); def_simp; eauto.
Qed.
End GluingInduction.
Variant glu_elem_bot i a Γ A M m : Prop :=
| glu_elem_bot_make : forall P El,
{{ Γ ⊢ M : A }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Γ ⊢ A ® P }} ->
{{ Dom m ≈ m ∈ per_bot }} ->
(forall Δ σ M', {{ Δ ⊢w σ : Γ }} -> {{ Rne m in length Δ ↘ M' }} -> {{ Δ ⊢ M[σ] ≈ M' : A[σ] }}) ->
{{ Γ ⊢ M : A ® m ∈ glu_elem_bot i a }}.
#[export]
Hint Constructors glu_elem_bot : mcltt.
Variant glu_elem_top i a Γ A M m : Prop :=
| glu_elem_top_make : forall P El,
{{ Γ ⊢ M : A }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Γ ⊢ A ® P }} ->
{{ Dom ⇓ a m ≈ ⇓ a m ∈ per_top }} ->
(forall Δ σ w, {{ Δ ⊢w σ : Γ }} -> {{ Rnf ⇓ a m in length Δ ↘ w }} -> {{ Δ ⊢ M[σ] ≈ w : A[σ] }}) ->
{{ Γ ⊢ M : A ® m ∈ glu_elem_top i a }}.
#[export]
Hint Constructors glu_elem_top : mcltt.
Variant glu_typ_top i a Γ A : Prop :=
| glu_typ_top_make :
{{ Γ ⊢ A : Type@i }} ->
{{ Dom a ≈ a ∈ per_top_typ }} ->
(forall Δ σ A', {{ Δ ⊢w σ : Γ }} -> {{ Rtyp a in length Δ ↘ A' }} -> {{ Δ ⊢ A[σ] ≈ A' : Type@i }}) ->
{{ Γ ⊢ A ® glu_typ_top i a }}.
#[export]
Hint Constructors glu_typ_top : mcltt.
Variant glu_rel_typ_with_sub i Δ A σ ρ : Prop :=
| mk_glu_rel_typ_with_sub :
`{ forall P El,
{{ ⟦ A ⟧ ρ ↘ a }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Δ ⊢ A[σ] ® P }} ->
glu_rel_typ_with_sub i Δ A σ ρ }.
Definition nil_glu_sub_pred : glu_sub_pred :=
fun Δ σ ρ => {{ Δ ⊢s σ : ⋅ }}.
Arguments nil_glu_sub_pred Δ σ ρ/.
Notation "'glu_exp_pred_args'" := (Tcons ctx (Tcons typ (Tcons exp (Tcons domain Tnil)))).
Notation "'glu_exp_pred'" := (predicate glu_exp_pred_args).
Notation "'glu_exp_pred_equivalence'" := (@predicate_equivalence glu_exp_pred_args) (only parsing).
Notation "Γ ⊢ M : A ® m ∈ R" := (R Γ A M m : (Prop : (Type : Type))) (in custom judg at level 80, Γ custom exp, M custom exp, A custom exp, m custom domain, R constr).
Notation "'glu_sub_pred_args'" := (Tcons ctx (Tcons sub (Tcons env Tnil))).
Notation "'glu_sub_pred'" := (predicate glu_sub_pred_args).
Notation "'glu_sub_pred_equivalence'" := (@predicate_equivalence glu_sub_pred_args) (only parsing).
Notation "Γ ⊢s σ ® ρ ∈ R" := ((R Γ σ ρ : Prop) : (Prop : (Type : Type))) (in custom judg at level 80, Γ custom exp, σ custom exp, ρ custom domain, R constr).
Notation "'DG' a ∈ R ↘ P ↘ El" := (R P El a : ((Prop : Type) : (Type : Type))) (in custom judg at level 90, a custom domain, R constr, P constr, El constr).
Notation "'EG' A ∈ R ↘ Sb " := (R Sb A : ((Prop : (Type : Type)) : (Type : Type))) (in custom judg at level 90, A custom exp, R constr, Sb constr).
Inductive glu_nat : ctx -> exp -> domain -> Prop :=
| glu_nat_zero :
`{ {{ Γ ⊢ M ≈ zero : ℕ }} ->
glu_nat Γ M d{{{ zero }}} }
| glu_nat_succ :
`{ {{ Γ ⊢ M ≈ succ M' : ℕ }} ->
glu_nat Γ M' m' ->
glu_nat Γ M d{{{ succ m' }}} }
| glu_nat_neut :
`{ per_bot m m ->
(forall {Δ σ M'}, {{ Δ ⊢w σ : Γ }} -> {{ Rne m in length Δ ↘ M' }} -> {{ Δ ⊢ M[σ] ≈ M' : ℕ }}) ->
glu_nat Γ M d{{{ ⇑ ℕ m }}} }.
#[export]
Hint Constructors glu_nat : mcltt.
Definition nat_glu_typ_pred i : glu_typ_pred := fun Γ A => {{ Γ ⊢ A ≈ ℕ : Type@i }}.
Arguments nat_glu_typ_pred i Γ A/.
Definition nat_glu_exp_pred i : glu_exp_pred := fun Γ A M m => {{ Γ ⊢ A ® nat_glu_typ_pred i }} /\ glu_nat Γ M m.
Arguments nat_glu_exp_pred i Γ A M m/.
Definition neut_glu_typ_pred i a : glu_typ_pred :=
fun Γ A => {{ Γ ⊢ A : Type@i }} /\
(forall Δ σ A', {{ Δ ⊢w σ : Γ }} -> {{ Rne a in length Δ ↘ A' }} -> {{ Δ ⊢ A[σ] ≈ A' : Type@i }}).
Arguments neut_glu_typ_pred i a Γ A/.
Variant neut_glu_exp_pred i a : glu_exp_pred :=
| mk_neut_glu_exp_pred :
`{ {{ Γ ⊢ A ® neut_glu_typ_pred i a }} ->
{{ Γ ⊢ M : A }} ->
{{ Dom m ≈ m ∈ per_bot }} ->
(forall Δ σ M', {{ Δ ⊢w σ : Γ }} ->
{{ Rne m in length Δ ↘ M' }} ->
{{ Δ ⊢ M[σ] ≈ M' : A[σ] }}) ->
{{ Γ ⊢ M : A ® ⇑ b m ∈ neut_glu_exp_pred i a }} }.
Variant pi_glu_typ_pred i
(IR : relation domain)
(IP : glu_typ_pred)
(IEl : glu_exp_pred)
(OP : forall c (equiv_c : {{ Dom c ≈ c ∈ IR }}), glu_typ_pred) : glu_typ_pred :=
| mk_pi_glu_typ_pred :
`{ {{ Γ ⊢ A ≈ Π IT OT : Type@i }} ->
{{ Γ ⊢ IT : Type@i }} ->
{{ Γ , IT ⊢ OT : Type@i }} ->
(forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
(forall Δ σ M m,
{{ Δ ⊢w σ : Γ }} ->
{{ Δ ⊢ M : IT[σ] ® m ∈ IEl }} ->
forall (equiv_m : {{ Dom m ≈ m ∈ IR }}),
{{ Δ ⊢ OT[σ,,M] ® OP _ equiv_m }}) ->
{{ Γ ⊢ A ® pi_glu_typ_pred i IR IP IEl OP }} }.
Variant pi_glu_exp_pred i
(IR : relation domain)
(IP : glu_typ_pred)
(IEl : glu_exp_pred)
(elem_rel : relation domain)
(OEl : forall c (equiv_c : {{ Dom c ≈ c ∈ IR }}), glu_exp_pred): glu_exp_pred :=
| mk_pi_glu_exp_pred :
`{ {{ Γ ⊢ M : A }} ->
{{ Dom m ≈ m ∈ elem_rel }} ->
{{ Γ ⊢ A ≈ Π IT OT : Type@i }} ->
{{ Γ ⊢ IT : Type@i }} ->
{{ Γ , IT ⊢ OT : Type@i }} ->
(forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
(forall Δ σ N n,
{{ Δ ⊢w σ : Γ }} ->
{{ Δ ⊢ N : IT[σ] ® n ∈ IEl }} ->
forall (equiv_n : {{ Dom n ≈ n ∈ IR }}),
exists mn, {{ $| m & n |↘ mn }} /\ {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl _ equiv_n }}) ->
{{ Γ ⊢ M : A ® m ∈ pi_glu_exp_pred i IR IP IEl elem_rel OEl }} }.
#[export]
Hint Constructors neut_glu_exp_pred pi_glu_typ_pred pi_glu_exp_pred : mcltt.
Definition univ_glu_typ_pred j i : glu_typ_pred := fun Γ A => {{ Γ ⊢ A ≈ Type@j : Type@i }}.
Arguments univ_glu_typ_pred j i Γ A/.
Transparent univ_glu_typ_pred.
Section Gluing.
Variable
(i : nat)
(glu_univ_typ_rec : forall {j}, j < i -> domain -> glu_typ_pred).
Definition univ_glu_exp_pred' {j} (lt_j_i : j < i) : glu_exp_pred :=
fun Γ A M m =>
{{ Γ ⊢ M : A }} /\
{{ Γ ⊢ A ≈ Type@j : Type@i }} /\
{{ Γ ⊢ M ® glu_univ_typ_rec lt_j_i m }}.
#[global]
Arguments univ_glu_exp_pred' {j} lt_j_i Γ A M m/.
Inductive glu_univ_elem_core : glu_typ_pred -> glu_exp_pred -> domain -> Prop :=
| glu_univ_elem_core_univ :
`{ forall typ_rel
el_rel
(lt_j_i : j < i),
typ_rel <∙> univ_glu_typ_pred j i ->
el_rel <∙> univ_glu_exp_pred' lt_j_i ->
{{ DG 𝕌@j ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
| glu_univ_elem_core_nat :
`{ forall typ_rel el_rel,
typ_rel <∙> nat_glu_typ_pred i ->
el_rel <∙> nat_glu_exp_pred i ->
{{ DG ℕ ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
| glu_univ_elem_core_pi :
`{ forall (in_rel : relation domain)
IP IEl
(OP : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_typ_pred)
(OEl : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_exp_pred)
typ_rel el_rel
(elem_rel : relation domain),
{{ DG a ∈ glu_univ_elem_core ↘ IP ↘ IEl }} ->
{{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
(forall {c} (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) b,
{{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
{{ DG b ∈ glu_univ_elem_core ↘ OP _ equiv_c ↘ OEl _ equiv_c }}) ->
{{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
typ_rel <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
el_rel <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
{{ DG Π a ρ B ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
| glu_univ_elem_core_neut :
`{ forall typ_rel el_rel,
{{ Dom b ≈ b ∈ per_bot }} ->
typ_rel <∙> neut_glu_typ_pred i b ->
el_rel <∙> neut_glu_exp_pred i b ->
{{ DG ⇑ a b ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }.
End Gluing.
#[export]
Hint Constructors glu_univ_elem_core : mcltt.
Equations glu_univ_elem (i : nat) : glu_typ_pred -> glu_exp_pred -> domain -> Prop by wf i :=
| i => glu_univ_elem_core i (fun j lt_j_i a Γ A => exists P El, {{ DG a ∈ glu_univ_elem j ↘ P ↘ El }} /\ {{ Γ ⊢ A ® P }}).
Definition glu_univ_typ (i : nat) (a : domain) : glu_typ_pred :=
fun Γ A => exists P El, {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} /\ {{ Γ ⊢ A ® P }}.
Arguments glu_univ_typ i a Γ A/.
Definition univ_glu_exp_pred j i : glu_exp_pred :=
fun Γ A M m =>
{{ Γ ⊢ M : A }} /\ {{ Γ ⊢ A ≈ Type@j : Type@i }} /\
{{ Γ ⊢ M ® glu_univ_typ j m }}.
Arguments univ_glu_exp_pred j i Γ A M m/.
Section GluingInduction.
Hypothesis
(motive : nat -> glu_typ_pred -> glu_exp_pred -> domain -> Prop)
(case_univ :
forall i j
(P : glu_typ_pred) (El : glu_exp_pred) (lt_j_i : j < i),
(forall P' El' a, {{ DG a ∈ glu_univ_elem j ↘ P' ↘ El' }} -> motive j P' El' a) ->
P <∙> univ_glu_typ_pred j i ->
El <∙> univ_glu_exp_pred j i ->
motive i P El d{{{ 𝕌 @ j }}})
(case_nat :
forall i (P : glu_typ_pred) (El : glu_exp_pred),
P <∙> nat_glu_typ_pred i ->
El <∙> nat_glu_exp_pred i ->
motive i P El d{{{ ℕ }}})
(case_pi :
forall i a B (ρ : env) (in_rel : relation domain) (IP : glu_typ_pred)
(IEl : glu_exp_pred) (OP : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_typ_pred)
(OEl : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_exp_pred) (P : glu_typ_pred) (El : glu_exp_pred)
(elem_rel : relation domain),
{{ DG a ∈ glu_univ_elem i ↘ IP ↘ IEl }} ->
motive i IP IEl a ->
{{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
(forall (c : domain) (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) (b : domain),
{{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
{{ DG b ∈ glu_univ_elem i ↘ OP c equiv_c ↘ OEl c equiv_c }}) ->
(forall (c : domain) (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) (b : domain),
{{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
motive i (OP c equiv_c) (OEl c equiv_c) b) ->
{{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
P <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
El <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
motive i P El d{{{ Π a ρ B }}})
(case_neut :
forall i b a
(P : glu_typ_pred)
(El : glu_exp_pred),
{{ Dom b ≈ b ∈ per_bot }} ->
P <∙> neut_glu_typ_pred i b ->
El <∙> neut_glu_exp_pred i b ->
motive i P El d{{{ ⇑ a b }}})
.
#[local]
Ltac def_simp := simp glu_univ_elem in *; mauto 3.
#[derive(equations=no, eliminator=no), tactic="def_simp"]
Equations glu_univ_elem_ind i P El a
(H : glu_univ_elem i P El a) : motive i P El a by wf i :=
| i, P, El, a, H =>
glu_univ_elem_core_ind
i
(fun j lt_j_i a Γ A => glu_univ_typ j a Γ A)
(motive i)
(fun j P' El' lt_j_i HP' HEl' =>
case_univ i j P' El' lt_j_i
(fun P'' El'' A H => glu_univ_elem_ind j P'' El'' A H)
HP'
HEl')
(case_nat i)
_ (* (case_pi i) *)
(case_neut i)
P El a
_.
Next Obligation.
eapply (case_pi i); def_simp; eauto.
Qed.
End GluingInduction.
Variant glu_elem_bot i a Γ A M m : Prop :=
| glu_elem_bot_make : forall P El,
{{ Γ ⊢ M : A }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Γ ⊢ A ® P }} ->
{{ Dom m ≈ m ∈ per_bot }} ->
(forall Δ σ M', {{ Δ ⊢w σ : Γ }} -> {{ Rne m in length Δ ↘ M' }} -> {{ Δ ⊢ M[σ] ≈ M' : A[σ] }}) ->
{{ Γ ⊢ M : A ® m ∈ glu_elem_bot i a }}.
#[export]
Hint Constructors glu_elem_bot : mcltt.
Variant glu_elem_top i a Γ A M m : Prop :=
| glu_elem_top_make : forall P El,
{{ Γ ⊢ M : A }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Γ ⊢ A ® P }} ->
{{ Dom ⇓ a m ≈ ⇓ a m ∈ per_top }} ->
(forall Δ σ w, {{ Δ ⊢w σ : Γ }} -> {{ Rnf ⇓ a m in length Δ ↘ w }} -> {{ Δ ⊢ M[σ] ≈ w : A[σ] }}) ->
{{ Γ ⊢ M : A ® m ∈ glu_elem_top i a }}.
#[export]
Hint Constructors glu_elem_top : mcltt.
Variant glu_typ_top i a Γ A : Prop :=
| glu_typ_top_make :
{{ Γ ⊢ A : Type@i }} ->
{{ Dom a ≈ a ∈ per_top_typ }} ->
(forall Δ σ A', {{ Δ ⊢w σ : Γ }} -> {{ Rtyp a in length Δ ↘ A' }} -> {{ Δ ⊢ A[σ] ≈ A' : Type@i }}) ->
{{ Γ ⊢ A ® glu_typ_top i a }}.
#[export]
Hint Constructors glu_typ_top : mcltt.
Variant glu_rel_typ_with_sub i Δ A σ ρ : Prop :=
| mk_glu_rel_typ_with_sub :
`{ forall P El,
{{ ⟦ A ⟧ ρ ↘ a }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Δ ⊢ A[σ] ® P }} ->
glu_rel_typ_with_sub i Δ A σ ρ }.
Definition nil_glu_sub_pred : glu_sub_pred :=
fun Δ σ ρ => {{ Δ ⊢s σ : ⋅ }}.
Arguments nil_glu_sub_pred Δ σ ρ/.
The parameters are ordered differently from the Agda version
so that we can return glu_sub_pred.
Variant cons_glu_sub_pred i Γ A (TSb : glu_sub_pred) : glu_sub_pred :=
| mk_cons_glu_sub_pred :
`{ forall P El,
{{ Δ ⊢s σ : Γ, A }} ->
{{ ⟦ A ⟧ ρ ↯ ↘ a }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
| mk_cons_glu_sub_pred :
`{ forall P El,
{{ Δ ⊢s σ : Γ, A }} ->
{{ ⟦ A ⟧ ρ ↯ ↘ a }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
Here we use {{{ A[Wk][σ] }}} instead of {{{ A[Wk∘σ] }}}
as syntactic judgement derived from that is
a more direct consequence of {{ Γ, A ⊢ #0 : A[Wk] }}
{{ Δ ⊢ #0[σ] : A[Wk][σ] ® ~(ρ 0) ∈ El }} ->
{{ Δ ⊢s Wk ∘ σ ® ρ ↯ ∈ TSb }} ->
{{ Δ ⊢s σ ® ρ ∈ cons_glu_sub_pred i Γ A TSb }} }.
Inductive glu_ctx_env : glu_sub_pred -> ctx -> Prop :=
| glu_ctx_env_nil :
`{ forall Sb,
Sb <∙> nil_glu_sub_pred ->
{{ EG ⋅ ∈ glu_ctx_env ↘ Sb }} }
| glu_ctx_env_cons :
`{ forall i TSb Sb,
{{ EG Γ ∈ glu_ctx_env ↘ TSb }} ->
{{ Γ ⊢ A : Type@i }} ->
(forall Δ σ ρ,
{{ Δ ⊢s σ ® ρ ∈ TSb }} ->
glu_rel_typ_with_sub i Δ A σ ρ) ->
Sb <∙> cons_glu_sub_pred i Γ A TSb ->
{{ EG Γ, A ∈ glu_ctx_env ↘ Sb }} }.
Variant glu_rel_exp_with_sub i Δ M A σ ρ : Prop :=
| mk_glu_rel_exp_with_sub :
`{ forall P El,
{{ ⟦ A ⟧ ρ ↘ a }} ->
{{ ⟦ M ⟧ ρ ↘ m }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Δ ⊢ M[σ] : A[σ] ® m ∈ El }} ->
glu_rel_exp_with_sub i Δ M A σ ρ }.
Variant glu_rel_sub_with_sub Δ τ (Sb : glu_sub_pred) σ ρ : Prop :=
| mk_glu_rel_sub_with_sub :
`{ {{ ⟦ τ ⟧s ρ ↘ ρ' }} ->
{{ Δ ⊢s τ ∘ σ ® ρ' ∈ Sb }} ->
glu_rel_sub_with_sub Δ τ Sb σ ρ}.
Definition glu_rel_ctx Γ : Prop := exists Sb, {{ EG Γ ∈ glu_ctx_env ↘ Sb }}.
Arguments glu_rel_ctx Γ/.
Definition glu_rel_exp Γ M A : Prop :=
exists Sb,
{{ EG Γ ∈ glu_ctx_env ↘ Sb }} /\
exists i,
forall Δ σ ρ,
{{ Δ ⊢s σ ® ρ ∈ Sb }} ->
glu_rel_exp_with_sub i Δ M A σ ρ.
Arguments glu_rel_exp Γ M A/.
Definition glu_rel_sub Γ τ Γ' : Prop :=
exists Sb Sb',
{{ EG Γ ∈ glu_ctx_env ↘ Sb }} /\
{{ EG Γ' ∈ glu_ctx_env ↘ Sb' }} /\
forall Δ σ ρ,
{{ Δ ⊢s σ ® ρ ∈ Sb }} ->
glu_rel_sub_with_sub Δ τ Sb' σ ρ.
Arguments glu_rel_sub Γ τ Γ'/.
Notation "⊩ Γ" := (glu_rel_ctx Γ) (in custom judg at level 80, Γ custom exp).
Notation "Γ ⊩ M : A" := (glu_rel_exp Γ M A) (in custom judg at level 80, Γ custom exp, M custom exp, A custom exp).
Notation "Γ ⊩s τ : Γ'" := (glu_rel_sub Γ τ Γ') (in custom judg at level 80, Γ custom exp, τ custom exp, Γ' custom exp).
{{ Δ ⊢s Wk ∘ σ ® ρ ↯ ∈ TSb }} ->
{{ Δ ⊢s σ ® ρ ∈ cons_glu_sub_pred i Γ A TSb }} }.
Inductive glu_ctx_env : glu_sub_pred -> ctx -> Prop :=
| glu_ctx_env_nil :
`{ forall Sb,
Sb <∙> nil_glu_sub_pred ->
{{ EG ⋅ ∈ glu_ctx_env ↘ Sb }} }
| glu_ctx_env_cons :
`{ forall i TSb Sb,
{{ EG Γ ∈ glu_ctx_env ↘ TSb }} ->
{{ Γ ⊢ A : Type@i }} ->
(forall Δ σ ρ,
{{ Δ ⊢s σ ® ρ ∈ TSb }} ->
glu_rel_typ_with_sub i Δ A σ ρ) ->
Sb <∙> cons_glu_sub_pred i Γ A TSb ->
{{ EG Γ, A ∈ glu_ctx_env ↘ Sb }} }.
Variant glu_rel_exp_with_sub i Δ M A σ ρ : Prop :=
| mk_glu_rel_exp_with_sub :
`{ forall P El,
{{ ⟦ A ⟧ ρ ↘ a }} ->
{{ ⟦ M ⟧ ρ ↘ m }} ->
{{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
{{ Δ ⊢ M[σ] : A[σ] ® m ∈ El }} ->
glu_rel_exp_with_sub i Δ M A σ ρ }.
Variant glu_rel_sub_with_sub Δ τ (Sb : glu_sub_pred) σ ρ : Prop :=
| mk_glu_rel_sub_with_sub :
`{ {{ ⟦ τ ⟧s ρ ↘ ρ' }} ->
{{ Δ ⊢s τ ∘ σ ® ρ' ∈ Sb }} ->
glu_rel_sub_with_sub Δ τ Sb σ ρ}.
Definition glu_rel_ctx Γ : Prop := exists Sb, {{ EG Γ ∈ glu_ctx_env ↘ Sb }}.
Arguments glu_rel_ctx Γ/.
Definition glu_rel_exp Γ M A : Prop :=
exists Sb,
{{ EG Γ ∈ glu_ctx_env ↘ Sb }} /\
exists i,
forall Δ σ ρ,
{{ Δ ⊢s σ ® ρ ∈ Sb }} ->
glu_rel_exp_with_sub i Δ M A σ ρ.
Arguments glu_rel_exp Γ M A/.
Definition glu_rel_sub Γ τ Γ' : Prop :=
exists Sb Sb',
{{ EG Γ ∈ glu_ctx_env ↘ Sb }} /\
{{ EG Γ' ∈ glu_ctx_env ↘ Sb' }} /\
forall Δ σ ρ,
{{ Δ ⊢s σ ® ρ ∈ Sb }} ->
glu_rel_sub_with_sub Δ τ Sb' σ ρ.
Arguments glu_rel_sub Γ τ Γ'/.
Notation "⊩ Γ" := (glu_rel_ctx Γ) (in custom judg at level 80, Γ custom exp).
Notation "Γ ⊩ M : A" := (glu_rel_exp Γ M A) (in custom judg at level 80, Γ custom exp, M custom exp, A custom exp).
Notation "Γ ⊩s τ : Γ'" := (glu_rel_sub Γ τ Γ') (in custom judg at level 80, Γ custom exp, τ custom exp, Γ' custom exp).