162 lines
4.8 KiB
Coq
162 lines
4.8 KiB
Coq
Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
|
|
Require Import fp_red.
|
|
From Hammer Require Import Tactics.
|
|
From Equations Require Import Equations.
|
|
Require Import ssreflect ssrbool.
|
|
Require Import Logic.PropExtensionality (propositional_extensionality).
|
|
From stdpp Require Import relations (rtc(..), rtc_subrel).
|
|
Import Psatz.
|
|
|
|
Definition ProdSpace {n} (PA : PTm n -> Prop)
|
|
(PF : PTm n -> (PTm n -> Prop) -> Prop) b : Prop :=
|
|
forall a PB, PA a -> PF a PB -> PB (PApp b a).
|
|
|
|
Definition SumSpace {n} (PA : PTm n -> Prop)
|
|
(PF : PTm n -> (PTm n -> Prop) -> Prop) t : Prop :=
|
|
SNe t \/ exists a b, rtc TRedSN t (PPair a b) /\ PA a /\ (forall PB, PF a PB -> PB b).
|
|
|
|
Definition BindSpace {n} p := if p is PPi then @ProdSpace n else SumSpace.
|
|
|
|
Reserved Notation "⟦ A ⟧ i ;; I ↘ S" (at level 70).
|
|
|
|
Inductive InterpExt {n} i (I : nat -> PTm n -> Prop) : PTm n -> (PTm n -> Prop) -> Prop :=
|
|
| InterpExt_Ne A :
|
|
SNe A ->
|
|
⟦ A ⟧ i ;; I ↘ SNe
|
|
| InterpExt_Bind p A B PA PF :
|
|
⟦ A ⟧ i ;; I ↘ PA ->
|
|
(forall a, PA a -> exists PB, PF a PB) ->
|
|
(forall a PB, PF a PB -> ⟦ subst_PTm (scons a VarPTm) B ⟧ i ;; I ↘ PB) ->
|
|
⟦ PBind p A B ⟧ i ;; I ↘ BindSpace p PA PF
|
|
|
|
| InterpExt_Univ j :
|
|
j < i ->
|
|
⟦ PUniv j ⟧ i ;; I ↘ (I j)
|
|
|
|
| InterpExt_Step A A0 PA :
|
|
TRedSN A A0 ->
|
|
⟦ A0 ⟧ i ;; I ↘ PA ->
|
|
⟦ A ⟧ i ;; I ↘ PA
|
|
where "⟦ A ⟧ i ;; I ↘ S" := (InterpExt i I A S).
|
|
|
|
|
|
Lemma InterpExt_Univ' n i I j (PF : PTm n -> Prop) :
|
|
PF = I j ->
|
|
j < i ->
|
|
⟦ PUniv j ⟧ i ;; I ↘ PF.
|
|
Proof. hauto lq:on ctrs:InterpExt. Qed.
|
|
|
|
Infix "<?" := Compare_dec.lt_dec (at level 60).
|
|
|
|
Equations InterpUnivN n (i : nat) : PTm n -> (PTm n -> Prop) -> Prop by wf i lt :=
|
|
InterpUnivN n i := @InterpExt n i
|
|
(fun j A =>
|
|
match j <? i with
|
|
| left _ => exists PA, InterpUnivN n j A PA
|
|
| right _ => False
|
|
end).
|
|
Arguments InterpUnivN {n}.
|
|
|
|
Lemma InterpExt_lt_impl n i I I' A (PA : PTm n -> Prop) :
|
|
(forall j, j < i -> I j = I' j) ->
|
|
⟦ A ⟧ i ;; I ↘ PA ->
|
|
⟦ A ⟧ i ;; I' ↘ PA.
|
|
Proof.
|
|
move => hI h.
|
|
elim : A PA /h.
|
|
- hauto q:on ctrs:InterpExt.
|
|
- hauto lq:on rew:off ctrs:InterpExt.
|
|
- hauto q:on ctrs:InterpExt.
|
|
- hauto lq:on ctrs:InterpExt.
|
|
Qed.
|
|
|
|
Lemma InterpExt_lt_eq n i I I' A (PA : PTm n -> Prop) :
|
|
(forall j, j < i -> I j = I' j) ->
|
|
⟦ A ⟧ i ;; I ↘ PA =
|
|
⟦ A ⟧ i ;; I' ↘ PA.
|
|
Proof.
|
|
move => hI. apply propositional_extensionality.
|
|
have : forall j, j < i -> I' j = I j by sfirstorder.
|
|
firstorder using InterpExt_lt_impl.
|
|
Qed.
|
|
|
|
Notation "⟦ A ⟧ i ↘ S" := (InterpUnivN i A S) (at level 70).
|
|
|
|
Lemma InterpUnivN_nolt n i :
|
|
@InterpUnivN n i = @InterpExt n i (fun j (A : PTm n) => exists PA, ⟦ A ⟧ j ↘ PA).
|
|
Proof.
|
|
simp InterpUnivN.
|
|
extensionality A. extensionality PA.
|
|
set I0 := (fun _ => _).
|
|
set I1 := (fun _ => _).
|
|
apply InterpExt_lt_eq.
|
|
hauto q:on.
|
|
Qed.
|
|
|
|
#[export]Hint Rewrite @InterpUnivN_nolt : InterpUniv.
|
|
|
|
Derive Dependent Inversion iinv with (forall n i I (A : PTm n) PA, InterpExt i I A PA) Sort Prop.
|
|
|
|
Lemma InterpExt_cumulative n i j I (A : PTm n) PA :
|
|
i <= j ->
|
|
⟦ A ⟧ i ;; I ↘ PA ->
|
|
⟦ A ⟧ j ;; I ↘ PA.
|
|
Proof.
|
|
move => h h0.
|
|
elim : A PA /h0;
|
|
hauto l:on ctrs:InterpExt solve+:(by lia).
|
|
Qed.
|
|
|
|
Lemma InterpUniv_cumulative n i (A : PTm n) PA :
|
|
⟦ A ⟧ i ↘ PA -> forall j, i <= j ->
|
|
⟦ A ⟧ j ↘ PA.
|
|
Proof.
|
|
hauto l:on rew:db:InterpUniv use:InterpExt_cumulative.
|
|
Qed.
|
|
|
|
Definition CR {n} (P : PTm n -> Prop) :=
|
|
(forall a, P a -> SN a) /\
|
|
(forall a, SNe a -> P a).
|
|
|
|
Lemma N_Exps n (a b : PTm n) :
|
|
rtc TRedSN a b ->
|
|
SN b ->
|
|
SN a.
|
|
Proof.
|
|
induction 1; eauto using N_Exp.
|
|
Qed.
|
|
|
|
Lemma adequacy_ext i n I A PA
|
|
(hI0 : forall j, j < i -> forall a b, (TRedSN a b) -> I j b -> I j a)
|
|
(hI : forall j, j < i -> CR (I j))
|
|
(h : ⟦ A : PTm n ⟧ i ;; I ↘ PA) :
|
|
CR PA /\ SN A.
|
|
Proof.
|
|
elim : A PA / h.
|
|
- hauto lq:on ctrs:SN unfold:CR.
|
|
- move => p A B PA PF hPA [ihA0 ihA1] hTot hRes ihPF.
|
|
have hb : PA PBot by hauto q:on ctrs:SNe.
|
|
have hb' : SN PBot by hauto q:on ctrs:SN, SNe.
|
|
rewrite /CR.
|
|
repeat split.
|
|
+ case : p =>//=.
|
|
* rewrite /ProdSpace.
|
|
qauto use:SN_AppInv unfold:CR.
|
|
* rewrite /SumSpace => a []; first by apply N_SNe.
|
|
move => [q0][q1]*.
|
|
have : SN q0 /\ SN q1 by hauto q:on.
|
|
hauto lq:on use:N_Pair,N_Exps.
|
|
+ move => a ha.
|
|
case : p=>/=.
|
|
* rewrite /ProdSpace => a0 *.
|
|
suff : SNe (PApp a a0) by sfirstorder.
|
|
hauto q:on use:N_App.
|
|
* rewrite /SumSpace. tauto.
|
|
+ apply N_Bind=>//=.
|
|
have : SN (PApp (PAbs B) PBot).
|
|
apply : N_Exp; eauto using N_β.
|
|
hauto lq:on.
|
|
qauto l:on use:SN_AppInv, SN_NoForbid.P_AbsInv.
|
|
- sfirstorder.
|
|
- hauto l:on ctrs:SN unfold:CR.
|
|
Qed.
|