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InstSem.v
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1505 lines (1261 loc) · 62.6 KB
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Require Import ModelTheory.
Export GenLemmas.
Import ZF Sat ZFuniv_real ZFcoc CC_Real SN_CC_Real SN_nat.
Lemma beta_nat_eq f n :
ext_fun N f ->
n ∈ N ->
app (lam (mkTY N cNAT) f) n == f n.
intros.
apply beta_eq; trivial.
red; intros.
rewrite El_def,eqNbot in H1; auto.
red; rewrite El_def,eqNbot; auto.
Qed.
Lemma app2_m2 m : morph2 (fun n x => app (app m n) x).
do 3 red; intros.
rewrite H,H0; reflexivity.
Qed.
Hint Resolve app2_m2.
Definition TT := prod props (fun p => prod p (fun p1 => p)).
Definition FF := prod props (fun p => p).
Definition DD := natrec TT (fun _ _ => FF).
Instance DD_morph : morph1 DD.
intros n n' eqn.
apply natrec_morph; auto with *.
do 2 red; reflexivity.
Qed.
Lemma DD_0 : DD zero == TT.
unfold DD; apply natrec_0.
Qed.
Lemma DD_S n : n ∈ N -> DD (succ n) == FF.
intros tyn.
apply natrec_S; trivial.
do 3 red; reflexivity.
Qed.
(*Instantiate the semantic of First Order Theory with Presburger*)
Module PresburgerSem <: TheorySem.
Include FOTheory_Cst.
Section SortAndEquation.
(*Presburger is the uni-sort theroy*)
Definition sort := Nat.
Lemma sort_not_kind : sort <> kind.
discriminate.
Qed.
Lemma sort_not_empty : exists t, forall e, typ e t sort.
exists Zero; intros. apply typ_0.
Qed.
(*The sort is closed, since it is first-order*)
Lemma sort_closed : forall i1 i2,
int sort i1 == int sort i2.
intros; simpl; reflexivity.
Qed.
(*Equation is encoded impredicatively*)
Definition EQ_term x y :=
Prod (Prod Nat prop) (Prod (App (Ref 0) (lift 1 x)) (App (Ref 1) (lift 2 y))).
Lemma EQ_term_typ : forall x y e,
typ e x Nat ->
typ e y Nat ->
typ e (EQ_term x y) prop.
intros; apply typ_prod; [right; trivial|left|].
apply typ_prod; [left; trivial|left; apply typ_N|apply typ_prop].
apply typ_prod; [right; trivial|right|].
setoid_replace prop with (subst (lift 1 x) prop) using relation eq_term at 2;
[|simpl; split; red; reflexivity].
apply typ_app with (V:=(lift 1 Nat));
[apply weakening; trivial| |discriminate|discriminate].
setoid_replace (Prod (lift 1 Nat) prop) with (lift 1 (Prod Nat prop)) using relation eq_term;
[apply typ_var; trivial|simpl; split; red; reflexivity].
setoid_replace prop with (subst (lift 2 y) prop) using relation eq_term at 2;
[|simpl; split; red; reflexivity].
apply typ_app with (V:=(lift 2 Nat));
[do 2 rewrite split_lift with (n:=1); do 2 apply weakening; trivial|
|discriminate|discriminate].
setoid_replace (Prod (lift 2 Nat) prop) with (lift 2 (Prod Nat prop)) using relation eq_term;
[apply typ_var; trivial|simpl; split; red; reflexivity].
Qed.
Definition P1 := lam (mkTY N cNAT) DD.
Lemma P1_real : [P1, Lc.K]\real prod (mkTY N cNAT) (fun _ => props).
assert (forall x, x ∈ N ->
natrec TT (fun _ _ => FF) x ∈ El props).
intros; change (El props) with ((fun _ => El props) x).
apply natrec_typ; [do 2 red; reflexivity|do 3 red; reflexivity|trivial| |intros].
apply impredicative_prod.
do 2 red; intros; apply prod_ext; [|do 2 red]; trivial.
intros; apply impredicative_prod; [do 2 red; reflexivity|trivial].
apply impredicative_prod; [do 2 red |]; trivial.
apply rprod_intro_lam.
do 2 red; intros; apply natrec_morph; [reflexivity|do 3 red; reflexivity|trivial].
do 2 red; reflexivity.
apply Lc.sn_abs; apply Lc.sn_var.
intros; destruct H0 as (Hx, Hu); unfold inX in Hx; rewrite El_def,eqNbot in Hx; split.
unfold inX; apply H; trivial.
unfold Lc.subst; simpl Lc.subst_rec. rewrite Real_sort; [clear H|apply H; trivial].
apply snSAT_intro;apply Lc.sn_abs; apply Lc.sn_lift; apply sat_sn in Hu; trivial.
Qed.
Lemma P1_ZERO : app P1 zero == TT.
unfold P1; rewrite beta_nat_eq.
apply DD_0.
do 2 red; intros; apply DD_morph; trivial.
apply zero_typ.
Qed.
Lemma P1_SUCC : forall n, n ∈ N -> app P1 (succ n) == FF.
unfold P1; intros; rewrite beta_nat_eq.
apply DD_S; trivial.
do 2 red; intros; apply DD_morph; trivial.
apply succ_typ; trivial.
Qed.
Lemma False_closed1 : forall n x t m' n' j,
n ∈ N ->
m' == zero ->
n' == succ n ->
(forall m, closed_pure_term (j m)) ->
~[x, tm t j]\real prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 m') (fun _ : X => app x0 n')).
intros n x t m' n' j Hn Hm' Hn' Hm Ht.
assert (prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 m') (fun _ : X => app x0 n')) ==
prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 zero) (fun _ : X => app x0 (succ n)))).
apply prod_ext; [reflexivity|do 2 red; intros].
apply prod_ext; [rewrite Hm'|do 2 red; intros; rewrite Hn']; rewrite H0; reflexivity.
rewrite H in Ht. clear m' n' Hm' Hn' H.
apply rprod_elim with (x:=P1) (u:=Lc.K) in Ht; [| |apply P1_real].
2: do 2 red; intros; apply prod_ext; [|do 2 red; intros]; rewrite H0; reflexivity.
apply rprod_elim with (x:=lam props (fun x => lam x (fun y => y)))
(u:=(Lc.Abs (Lc.Abs (Lc.Ref 0)))) in Ht; [|do 2 red; reflexivity|].
set (prf:=Lc.App (Lc.App (tm t j) Lc.K) (Lc.Abs (Lc.Abs (Lc.Ref 0)))) in Ht.
assert (forall S, inSAT (Lc.App prf (Lc.Abs (Lc.Ref 0))) S).
intros; assert ([mkProp S, (Lc.Abs (Lc.Ref 0))] \real props).
assert (mkProp S ∈ El props).
apply mkProp_intro.
split; trivial.
rewrite Real_sort; trivial.
apply snSAT_intro;apply Lc.sn_abs; apply Lc.sn_var.
assert ([app (app x P1) (lam props (fun x => (lam x (fun y => y)))), prf] \real
app P1 (succ n) ->
[app (app x P1) (lam props (fun x => (lam x (fun y => y)))), prf] \real
prod props (fun p => p)).
apply real_morph; [|rewrite P1_SUCC; trivial|]; reflexivity.
apply H0 in Ht; clear H0.
assert (H2 := @rprod_elim props (app (app x P1)
(lam props (fun x => (lam x (fun y => y))))) (mkProp S)
(fun P=>P) prf (Lc.Abs (Lc.Ref 0))).
destruct H2; [do 2 red| | |]; trivial.
rewrite Real_mkProp in H1; trivial.
apply rprod_elim with (x:=(mkProp S)) (u:=(Lc.Abs (Lc.Ref 0))) in Ht;
[|do 2 red|]; trivial.
unfold inX in H0; rewrite El_mkProp in H0; apply singl_elim in H0; trivial.
destruct (neutral_not_closed _ H). inversion_clear H0.
inversion_clear H1.
inversion_clear H0.
apply tm_closed in H1. unfold closed_pure_term in Hm. apply H1. intros n0 HF.
apply Hm with (m:=n0) (k:=x0); trivial.
inversion_clear H1; inversion_clear H0; inversion_clear H1.
inversion_clear H0; inversion_clear H1; inversion H0.
inversion_clear H1. inversion_clear H0.
assert ([lam props (fun x => lam x (fun y => y)), Lc.Abs (Lc.Abs (Lc.Ref 0))] \real
prod props (fun p => prod p (fun x => p)) ->
[lam props (fun x => lam x (fun y => y)), Lc.Abs (Lc.Abs (Lc.Ref 0))]\real app P1 zero).
apply real_morph; [|rewrite P1_ZERO; trivial|]; reflexivity.
apply H; clear H.
apply rprod_intro_lam; [do 2 red; intros; apply lam_ext; [|do 2 red; intros]; trivial
| do 2 red; intros; apply prod_ext; [|do 2 red; intros]; trivial
| apply Lc.sn_abs; apply Lc.sn_var
| unfold Lc.subst; simpl Lc.subst_rec; intros].
apply rprod_intro_lam; [do 2 red; trivial|do 2 red; reflexivity|apply Lc.sn_var|].
unfold Lc.subst; simpl Lc.subst_rec; intros; rewrite Lc.lift0; trivial.
Qed.
Definition P2 := lam (mkTY N cNAT) (natrec FF (fun _ _ => TT)).
Lemma P2_real : [P2, Lc.K]\real prod (mkTY N cNAT) (fun _ : X => props).
assert (forall x, x ∈ N ->
natrec FF (fun _ _ => TT) x ∈ El props).
intros; change (El props) with ((fun _ => El props) x).
apply natrec_typ; [do 2 red; reflexivity|do 3 red; reflexivity|trivial| |intros].
apply impredicative_prod; [do 2 red |]; trivial.
apply impredicative_prod; [do 2 red|]; intros.
apply prod_ext; [|do 2 red]; trivial.
apply impredicative_prod; [do 2 red; reflexivity|trivial].
apply rprod_intro_lam.
do 2 red; intros; apply natrec_morph; [reflexivity|do 3 red; reflexivity|trivial].
do 2 red; reflexivity.
apply Lc.sn_abs; apply Lc.sn_var.
intros; destruct H0 as (Hx, Hu); unfold inX in Hx; rewrite El_def,eqNbot in Hx; split.
unfold inX; apply H; trivial.
unfold Lc.subst; simpl Lc.subst_rec. rewrite Real_sort; [clear H|apply H; trivial].
apply snSAT_intro;apply Lc.sn_abs; apply Lc.sn_lift; apply sat_sn in Hu; trivial.
Qed.
Lemma P2_SUCC : forall n, n ∈ N ->
app P2 (succ n) == TT.
intros; unfold P2; rewrite beta_nat_eq.
rewrite natrec_S; [reflexivity|do 3 red; reflexivity|trivial].
do 2 red; intros. apply natrec_morph; [reflexivity|do 3 red; intros; reflexivity|trivial].
apply succ_typ; trivial.
Qed.
Lemma P2_ZERO : app P2 zero == FF.
unfold P2; rewrite beta_nat_eq.
apply natrec_0.
do 2 red; intros; apply natrec_morph; [reflexivity|do 3 red; reflexivity|trivial].
apply zero_typ.
Qed.
Lemma False_closed2 : forall n x t m1 m2 j,
n ∈ N ->
m1 == succ n ->
m2 == zero ->
(forall m, closed_pure_term (j m)) ->
~[x, tm t j]\real prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 m1) (fun _ : X => app x0 m2)).
intros n x t m1 m2 j Hn Hm1 Hm2 Hm Ht.
assert ([x, tm t j]\real
prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 m1) (fun _ : X => app x0 m2)) ->
[x, tm t j]\real
prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 (succ n)) (fun _ : X => app x0 zero))).
apply real_morph; [reflexivity| |reflexivity].
apply prod_ext; [reflexivity|do 2 red; intros].
apply prod_ext; [rewrite Hm1|do 2 red; intros; rewrite Hm2]; rewrite H0; reflexivity.
apply H in Ht; clear H.
apply rprod_elim with (x:=P2) (u:=Lc.K) in Ht; [| |apply P2_real].
2: do 2 red; intros; apply prod_ext; [|do 2 red; intros]; rewrite H0; reflexivity.
apply rprod_elim with (x:=lam props (fun x => lam x (fun y => y)))
(u:=(Lc.Abs (Lc.Abs (Lc.Ref 0)))) in Ht; [|do 2 red; reflexivity|].
set (prf:=Lc.App (Lc.App (tm t j) Lc.K) (Lc.Abs (Lc.Abs (Lc.Ref 0)))) in Ht.
assert (forall S, inSAT (Lc.App prf (Lc.Abs (Lc.Ref 0))) S).
intros; assert ([mkProp S, (Lc.Abs (Lc.Ref 0))] \real props).
assert (mkProp S ∈ El props).
apply mkProp_intro.
split; trivial.
rewrite Real_sort; trivial.
apply snSAT_intro;apply Lc.sn_abs; apply Lc.sn_var.
assert ([app (app x P2) (lam props (fun x => (lam x (fun y => y)))), prf] \real
app P2 zero ->
[app (app x P2) (lam props (fun x => (lam x (fun y => y)))), prf] \real
prod props (fun p => p)).
apply real_morph; [|rewrite P2_ZERO; trivial|]; reflexivity.
apply H0 in Ht; clear H0.
assert (H2 := @rprod_elim props (app (app x P2)
(lam props (fun x => (lam x (fun y => y))))) (mkProp S)
(fun P=>P) prf (Lc.Abs (Lc.Ref 0))).
destruct H2; [do 2 red| | |]; trivial.
rewrite Real_mkProp in H1; trivial.
apply rprod_elim with (x:=(mkProp S)) (u:=(Lc.Abs (Lc.Ref 0))) in Ht;
[|do 2 red|]; trivial.
unfold inX in H0; rewrite El_mkProp in H0; apply singl_elim in H0; trivial.
destruct (neutral_not_closed _ H). inversion_clear H0.
inversion_clear H1.
inversion_clear H0.
apply tm_closed in H1. unfold closed_pure_term in Hm. apply H1. intros n0 HF.
apply Hm with (m:=n0) (k:=x0); trivial.
inversion_clear H1; inversion_clear H0; inversion_clear H1.
inversion_clear H0; inversion_clear H1; inversion H0.
inversion_clear H1. inversion_clear H0.
assert ([lam props (fun x => lam x (fun y => y)), Lc.Abs (Lc.Abs (Lc.Ref 0))] \real
prod props (fun p => prod p (fun x => p)) ->
[lam props (fun x => lam x (fun y => y)), Lc.Abs (Lc.Abs (Lc.Ref 0))]\real app P2 (succ n)).
apply real_morph; [|rewrite P2_SUCC; trivial|]; reflexivity.
apply H; clear H.
apply rprod_intro_lam; [do 2 red; intros; apply lam_ext; [|do 2 red; intros]; trivial
| do 2 red; intros; apply prod_ext; [|do 2 red; intros]; trivial
| apply Lc.sn_abs; apply Lc.sn_var
| unfold Lc.subst; simpl Lc.subst_rec; intros].
apply rprod_intro_lam; [do 2 red; trivial|do 2 red; reflexivity|apply Lc.sn_var|].
unfold Lc.subst; simpl Lc.subst_rec; intros; rewrite Lc.lift0; trivial.
Qed.
Definition P3 x0 := (lam (mkTY N cNAT) (natrec FF (fun n _ => app x0 n))).
Lemma P3_real : forall x0 u,
[x0, u]\real prod (mkTY N cNAT) (fun _ => props) ->
[P3 x0, u]\real prod (mkTY N cNAT) (fun _ => props).
intros.
assert (forall x, x ∈ N ->
natrec FF (fun n _ : set => app x0 n) x ∈ El props).
intros; change (El props) with ((fun _ => El props) x).
apply natrec_typ; [do 2 red; reflexivity|do 3 red; intros; rewrite H1; reflexivity
|trivial| |intros].
apply impredicative_prod; [do 2 red |]; trivial.
specialize (inSAT_n k H1); intros H3. destruct H3 as (x1, (H3, _)).
apply rprod_elim with (x:=k) (u:=x1) in H; [|do 2 red; intros; reflexivity|].
destruct H as (H, _); unfold inX in H; trivial.
split; [unfold inX; rewrite El_def,eqNbot|]; trivial.
rewrite Real_def; auto.
intros; apply cNAT_morph; trivial.
apply rprod_intro_sn; [|do 2 red; reflexivity|apply real_sn in H; trivial|].
do 2 red; intros; apply natrec_morph; [reflexivity
|do 3 red; intros; rewrite H3; reflexivity|trivial].
intros. assert (x ∈ N).
destruct H1 as (H1, _). unfold inX in H1; rewrite El_def,eqNbot in H1; trivial.
apply rprod_elim with (x:=x) (u:=u0) in H; [apply H0 in H2|do 2 red; reflexivity|trivial].
split; [unfold inX|rewrite Real_sort; [apply real_sn in H|]]; trivial.
Qed.
Lemma P3_SUCC : forall n x0,
n ∈ N ->
app (P3 x0) (succ n) == app x0 n.
intros; unfold P3; rewrite beta_nat_eq.
rewrite natrec_S; [reflexivity|do 3 red; intros; rewrite H0; reflexivity|trivial].
do 2 red; intros. apply natrec_morph;
[reflexivity|do 3 red; intros; rewrite H2; reflexivity|trivial].
apply succ_typ; trivial.
Qed.
Lemma eq_SUCC_eq : forall m n x y,
m ∈ N ->
n ∈ N ->
[x, y]\real prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 (succ n)) (fun _ : X => app x0 (succ m))) ->
[x, y]\real prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 n) (fun _ : X => app x0 m)).
intros m n x y Hm Hn HS.
assert (forall m n, m ∈ N -> n ∈ N ->
prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 n) (fun _ : X => app x0 m)) ∈ El props).
intros. apply impredicative_prod.
do 2 red; intros. apply prod_ext; [|do 2 red; intros]; rewrite H2; reflexivity.
intros; apply impredicative_prod.
do 2 red; intros; reflexivity.
apply prod_elim with (x:=m0) in H1; trivial.
do 2 red; intros; reflexivity.
rewrite El_def,eqNbot; trivial.
assert (prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 (succ n)) (fun _ : X => app x0 (succ m))) ∈ El props).
apply H; apply succ_typ; trivial.
assert (prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 n) (fun _ : X => app x0 m)) ∈ El props).
apply H; trivial.
(*rewrite El_props_def in H0, H1. destruct H0, H1.*)
assert ((lam (prod (mkTY N cNAT) (fun _ => props))
(fun x0 => lam (app x0 n) (fun x1 => app (app x (P3 x0)) x1))) ∈
El (prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 n) (fun _ : X => app x0 m)))).
apply prod_intro.
do 2 red; intros. apply lam_ext; [rewrite H3; reflexivity|do 2 red; intros; rewrite H5].
apply app_ext; [|reflexivity].
apply app_ext; [reflexivity|unfold P3].
apply lam_ext; [reflexivity|do 2 red; intros].
apply natrec_morph;
[reflexivity|do 2 red; intros; rewrite H3; rewrite H8; reflexivity|trivial].
do 2 red; intros. apply prod_ext; [|do 2 red; intros]; rewrite H3; reflexivity.
intros. apply prod_intro.
do 2 red; intros. rewrite H4; reflexivity.
do 2 red; intros. reflexivity.
intros. destruct HS as (HS, _); unfold inX in HS.
pose (x2:=x0(*empty*)).
pose (x3:=x1(*empty*)).
assert (P3 x2 ∈ El (prod (mkTY N cNAT) (fun _ : X => props))).
unfold P3. apply prod_intro.
do 2 red; intros.
apply natrec_morph; [reflexivity|do 3 red; intros; rewrite H6; reflexivity|trivial].
do 2 red; intros; reflexivity.
intros. change (El props) with ((fun _ => El props) x4). apply natrec_typ; intros.
do 2 red; intros; reflexivity.
do 3 red; intros. rewrite H5; reflexivity.
rewrite El_def,eqNbot in H4; trivial.
apply impredicative_prod; [do 2 red|]; trivial.
apply prod_elim with (x:=k) in H2;
[|do 2 red; intros; reflexivity|rewrite El_def,eqNbot]; trivial.
apply prod_elim with (x:=(P3 x2)) in HS; [|do 2 red; intros|trivial].
2 : apply prod_ext; [|do 2 red; intros]; rewrite H6; reflexivity.
assert (El (prod (app (P3 x2) (succ n)) (fun _ : X => app (P3 x2) (succ m))) ==
El (prod (app x2 n) (fun _ : X => app x2 m))).
apply El_morph; apply prod_ext; [|do 2 red; intros]; rewrite P3_SUCC; trivial; reflexivity.
rewrite H5 in HS; clear H5.
apply prod_elim with (x:=x3) in HS; trivial.
do 2 red; intros; reflexivity.
assert (x == (lam (prod (mkTY N cNAT) (fun _ => props))
(fun x0 => lam (app x0 n) (fun x1 => app (app x (P3 x0)) x1)))).
destruct HS as (HS, _); unfold inX in H1.
red in HS; rewrite El_props_true with (1:=H0) in HS.
rewrite El_props_true with (1:=H1) in H2.
apply singl_elim in HS; apply singl_elim in H2.
rewrite HS,H2; reflexivity.
(* rewrite H0 in HS; rewrite H1 in H2. rewrite El_mkProp in HS, H2.
apply singl_elim in HS; apply singl_elim in H2. rewrite HS, H2; reflexivity.*)
assert ([lam (prod (mkTY N cNAT) (fun _ : set => props))
(fun x0 : set => lam (app x0 n) (fun x1 => app (app x (P3 x0)) x1)), y]\real
prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x2 : X => prod (app x2 n) (fun _ : X => app x2 m)) ->
[x, y]\real prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x2 : X => prod (app x2 n) (fun _ : X => app x2 m))).
apply real_morph; [trivial| |]; reflexivity.
apply H4. clear H0 (*x0*) H (*x1*) H1 H2 H3 H4.
apply rprod_intro_sn.
do 2 red; intros. apply lam_ext; [rewrite H0; reflexivity|do 2 red; intros].
apply app_ext; trivial.
apply app_ext; [reflexivity|].
unfold P3. apply lam_ext; [reflexivity|do 2 red; intros].
apply natrec_morph; [reflexivity|do 3 red; intros; rewrite H0, H5; reflexivity|trivial].
do 2 red; intros. apply prod_ext; [|do 2 red; intros]; rewrite H0; reflexivity.
apply real_sn in HS; trivial.
intros. apply rprod_intro_sn.
do 2 red; intros. rewrite H1; reflexivity.
do 2 red; intros. reflexivity.
apply rprod_elim with (3:=H) in HS.
apply real_sn in HS; trivial.
do 2 red; intros. apply prod_ext; [|do 2 red; intros]; rewrite H1; reflexivity.
intros. apply rprod_elim with (x:=P3 x0) (u:=u) in HS; [| |apply P3_real; trivial].
2 : do 2 red; intros; apply prod_ext; [|do 2 red; intros]; rewrite H2; reflexivity.
assert ([app x (P3 x0), GenRealSN.Lc.App y u]\real
prod (app (P3 x0) (succ n)) (fun _ : X => app (P3 x0) (succ m)) ->
[app x (P3 x0), GenRealSN.Lc.App y u]\real
prod (app x0 n) (fun _ : X => app x0 m)).
apply real_morph; [reflexivity| |reflexivity].
apply prod_ext; [|do 2 red; intros]; rewrite <- P3_SUCC; trivial; reflexivity.
apply H1 in HS. clear H1.
apply rprod_elim with (x:=x1) (u:=u0) in HS; [|do 2 red; reflexivity|]; trivial.
Qed.
Fixpoint const_env n :=
match n with
| 0 => nil
| S m => sort :: (const_env m)
end.
Lemma const_env0 : const_env 0 = nil.
simpl; trivial.
Qed.
Lemma const_envS : forall n, const_env (S n) = sort :: (const_env n).
simpl; trivial.
Qed.
Lemma const_env_intro : forall m n,
(m < n)%nat -> nth_error (const_env n) m = value Nat.
induction m; destruct n; simpl; intros; [|trivial| |apply IHm]; omega.
Qed.
Lemma const_env_elim : forall m n t,
nth_error (const_env n) m = value t ->
(m < n)%nat /\ t = Nat.
induction m; destruct n; simpl; intros; [discriminate| |discriminate|].
split; [omega|injection H; intros; subst t]; trivial.
apply IHm in H. destruct H; split; [omega|trivial].
Qed.
Lemma const_env_j : forall n i,
(forall m, (m < n)%nat -> i m ∈ N) ->
exists j, val_ok (const_env n) i j /\ (forall m, closed_pure_term (j m)).
induction n; intros.
exists (fun _ => Lc.Abs (Lc.Ref 0)); split; [red|]; intros.
destruct n; simpl in H; discriminate.
unfold closed_pure_term; intros k HF; inversion_clear HF. inversion H0.
specialize IHn with (i:=V.shift 1 i).
assert (forall m : nat, (m < n)%nat -> V.shift 1 i m ∈ N).
intros. unfold V.shift. apply H. omega.
specialize IHn with (1:=H0); clear H0.
destruct IHn as (j, (Hvalm, Hclsd)).
assert (i 0 ∈ N) as Hcons by (apply H; omega).
specialize inSAT_n with (1:=Hcons). intros HinSAT_n.
destruct HinSAT_n as (t, (HinSAT, Hclsdt)).
exists (I.cons t j); split.
assert (val_ok (const_env (S n)) (V.cons (i 0) (V.shift 1 i)) (I.cons t j) ->
val_ok (const_env (S n)) i (I.cons t j)).
intros H'; red in H' |- *; intros. apply H' in H0; revert H0.
apply in_int_morph; try reflexivity.
rewrite V.cons_ext with (i':=i); reflexivity.
apply H0; clear H0.
apply vcons_add_var; [|
split; [unfold inX; rewrite ElNat_eq|rewrite RealNat_eq]|discriminate]; trivial.
intros m; destruct m; simpl; trivial.
Qed.
Lemma EQ_term_eq_typ : forall n x y t,
typ (const_env n) x Nat ->
typ (const_env n) y Nat ->
typ (const_env n) t (EQ_term x y) ->
eq_typ (const_env n) x y.
do 2 red; intros n x y t Hx Hy Ht i j' Hok'.
assert (forall m, (m < n)%nat -> i m ∈ N).
intros; red in Hok'. apply const_env_intro in H. apply Hok' in H.
apply in_int_not_kind in H; [|discriminate]. destruct H as (H, _).
unfold inX in H. simpl int in H. rewrite El_def,eqNbot in H; trivial.
apply const_env_j in H. destruct H as (j, (Hok, Hclsd)). clear Hok' j'.
apply red_typ with (1:=Hok) in Ht; [destruct Ht as (_, Ht)|discriminate].
assert ([int t i, tm t j] \real int (EQ_term x y) i ->
[int t i, tm t j] \real prod (prod (mkTY N cNAT) (fun _ : X => props))
(fun x0 : X => prod (app x0 (int x i)) (fun x : X => app x0 (int y i)))).
simpl int; apply real_morph; [reflexivity | |reflexivity].
apply prod_ext; [reflexivity | do 2 red; intros].
apply prod_ext; [|do 2 red; intros]; rewrite H0; [|rewrite split_lift];
repeat rewrite int_cons_lift_eq; reflexivity.
apply H in Ht; clear H.
apply red_typ with (1:=Hok) in Hx; [|discriminate].
apply red_typ with (1:=Hok) in Hy; [|discriminate].
destruct Hx as (_, (Hx, _)). destruct Hy as (_, (Hy, _)).
unfold inX in Hx, Hy; clear Hok.
simpl int in Hx, Hy; rewrite El_def,eqNbot in Hx, Hy.
set (int_x := int x i) in *. clearbody int_x.
set (int_y := int y i) in *. clearbody int_y.
clear x y.
revert int_y Hy Ht; pattern int_x; apply N_ind; [| | |exact Hx]; intros.
rewrite <- H0. apply H1; [trivial|].
revert Ht; apply real_morph; [reflexivity| |reflexivity].
apply prod_ext; [reflexivity|do 2 red; intros].
apply prod_ext; [rewrite H0|do 2 red; intros]; rewrite H3; reflexivity.
revert Ht; pattern int_y; apply N_ind; [|reflexivity| |exact Hy]; intros.
rewrite <- H0; apply H1; revert Ht; apply real_morph; [reflexivity| |reflexivity].
apply prod_ext; [reflexivity|do 2 red; intros].
apply prod_ext; [|do 2 red; intros; rewrite H0]; rewrite H3; reflexivity.
apply False_closed1 with (n:=n0) in Ht;
[contradiction|trivial|reflexivity|reflexivity|trivial].
revert Ht; pattern int_y; apply N_ind; [| | |exact Hy]; intros.
rewrite <- H2; apply H3; revert Ht; apply real_morph; [reflexivity| |reflexivity].
apply prod_ext; [reflexivity|do 2 red; intros].
apply prod_ext; [|do 2 red; intros; rewrite H2]; rewrite H5; reflexivity.
apply False_closed2 with (n:=n0) in Ht;
[contradiction|trivial|reflexivity|reflexivity|trivial].
apply succ_morph. apply H0; [trivial|]. apply eq_SUCC_eq; trivial.
Qed.
End SortAndEquation.
Section FallAndExst.
Definition Fall A := Prod Nat A.
Lemma Fall_typ : forall e A,
typ (Nat::e) A prop ->
typ e (Fall A) prop.
intros; apply typ_prod; [right|unfold typs; left; apply typ_N|];trivial.
Qed.
Lemma Fall_intro : forall e t B,
typ (Nat::e) B prop ->
typ (Nat::e) t B ->
typ e (Abs Nat t) (Fall B).
intros e t B HB Ht i j Hok'.
assert (val_ok (Nat::e) (V.cons zero i) (I.cons ZE j)) as Hok.
apply vcons_add_var; [trivial| |discriminate].
generalize (typ_0 Hok'); intros typ0. apply typ0.
generalize HB; intros HSB.
apply red_typ with (1:=Hok) in HSB; [destruct HSB as (HSB, _)|discriminate].
clear Hok; revert i j Hok'; fold (typ e (Abs Nat t) (Fall B)).
apply typ_abs; [left; apply typ_N| |]; trivial.
Qed.
Lemma Fall_elim : forall e t u B,
typ (Nat::e) B prop ->
typ e t (Fall B) ->
typ e u Nat ->
typ e (App t u) (subst u B).
red; intros e t u B HB Ht Hu i j Hok.
assert (val_ok (Nat::e) (V.cons zero i) (I.cons ZE j)) as Hok'.
apply vcons_add_var; [trivial| |discriminate].
generalize (typ_0 Hok); intros typ0. apply typ0.
generalize HB; intros HSB.
apply red_typ with (1:=Hok') in HSB; [destruct HSB as (HSB, _)|discriminate].
clear Hok'; revert i j Hok; fold (typ e (App t u) (subst u B)).
apply typ_app with (V:=Nat); [| |discriminate|]; trivial.
Qed.
(*Exst for exst*)
Definition Exst A := Prod prop (
Prod (Prod Nat (Prod (subst (Ref 0) (lift_rec 2 1 A)) (Ref 2))) (Ref 1)).
Lemma Exst_typ : forall e A,
typ (Nat::e) A prop ->
typ e (Exst A) prop.
intros e A HA; unfold Exst.
apply typ_prod; [right; trivial|left; apply typ_prop|].
setoid_replace Nat with (lift 1 Nat) using relation eq_term;[|
simpl; split; red; reflexivity].
apply typ_prod; [right|right
|setoid_replace prop with (lift 2 prop) using relation eq_term at 2;
[apply typ_var|simpl; split; red; reflexivity]]; trivial.
apply typ_prod; [right; trivial
|left; setoid_replace kind with (lift 1 kind) using relation eq_term;
[apply weakening; apply typ_N|simpl; split; red; reflexivity]|].
apply typ_prod; [right|right
|setoid_replace prop with (lift 3 prop) using relation eq_term at 2;
[apply typ_var|simpl; split; red; reflexivity]]; trivial.
rewrite subst0_lift.
setoid_replace prop with (lift_rec 1 1 prop) using relation eq_term at 2;
[apply weakening_bind; trivial|simpl; split; red; reflexivity].
Qed.
Lemma Exst_intro : forall e A p a,
typ (Nat::e) A prop ->
typ e a Nat ->
typ e p (subst a A) ->
exists t, typ e t (Exst A).
intros e A p a HA Ha Hp.
exists (Abs prop (Abs (Prod Nat (Prod (subst (Ref 0) (lift_rec 2 1 A)) (Ref 2)))
(App (App (Ref 0) (lift 2 a)) (lift 2 p)))).
red; intros i j Hok.
assert (val_ok (Nat :: e) (V.cons zero i) (I.cons ZE j)).
apply vcons_add_var; [trivial| |discriminate].
generalize (typ_0 Hok); intros typ0. apply typ0.
generalize HA; intros HSA.
apply red_typ with (1:=H) in HSA; [destruct HSA as (HSA, _)|discriminate].
clear H; revert i j Hok;
fold (typ e (Abs prop (Abs (Prod Nat (Prod (subst (Ref 0) (lift_rec 2 1 A)) (Ref 2)))
(App (App (Ref 0) (lift 2 a)) (lift 2 p)))) (Exst A)).
apply typ_abs; [left; apply typ_prop| |discriminate].
apply typ_abs; [right| |discriminate].
setoid_replace Nat with (lift 1 Nat) using relation eq_term;[|
simpl; split; red; reflexivity].
apply typ_prod; [right; trivial|left;
setoid_replace (lift 1 Nat) with Nat using relation eq_term;[apply typ_N|
simpl; split; red; reflexivity]|].
apply typ_prod; [right; trivial|right|].
setoid_replace prop with (lift_rec 1 1 prop) using relation eq_term at 2;
[|simpl; split; red; reflexivity].
rewrite subst0_lift; apply weakening_bind; trivial.
setoid_replace prop with (lift 3 prop) using relation eq_term at 2;
[apply typ_var; trivial|simpl; split; red; reflexivity].
setoid_replace (Ref 1) with (subst (lift 2 p) (Ref 2)) using relation eq_term; [
|unfold subst; rewrite red_sigma_var_gt; [reflexivity|omega]].
apply typ_app with (V:=(lift 2 (subst a A)));
[| |destruct A; [discriminate|trivial]|discriminate].
do 2 rewrite split_lift with (n:=1). do 2 apply weakening; trivial.
assert (eq_term (Prod (lift 2 (subst a A)) (Ref 2))
(subst (lift 2 a) (Prod (subst (Ref 0) (lift_rec 3 1 A)) (Ref 3)))).
unfold subst at 2. rewrite red_sigma_prod. rewrite red_sigma_var_gt; [|omega].
apply Prod_morph; [rewrite subst0_lift|reflexivity].
apply eq_term_intro; intros.
unfold lift, subst. rewrite int_lift_rec_eq.
do 2 rewrite int_subst_rec_eq. do 2 rewrite int_lift_rec_eq. do 4 rewrite V.lams0.
rewrite <- V.cons_lams; [rewrite V.lams0|do 2 red; intros; rewrite H]; reflexivity.
unfold lift, subst. rewrite tm_lift_rec_eq.
do 2 rewrite tm_subst_rec_eq. do 2 rewrite tm_lift_rec_eq. do 4 rewrite I.lams0.
rewrite <- I.cons_lams; [rewrite I.lams0|do 2 red; intros; rewrite H]; reflexivity.
destruct A; simpl; trivial.
rewrite H; clear H.
apply typ_app with (V:=(lift 2 Nat)); [| |discriminate|discriminate].
do 2 rewrite split_lift with (n:=1). do 2 apply weakening; trivial.
setoid_replace (lift 2 Nat) with (lift 1 Nat) using relation eq_term;
[|simpl; split; red; reflexivity].
assert (eq_term (Prod (subst (Ref 0) (lift_rec 3 1 A)) (Ref 3))
(lift_rec 1 1 (Prod (subst (Ref 0) (lift_rec 2 1 A)) (Ref 2)))).
rewrite red_lift_prod. rewrite eq_term_lift_ref_fv by omega.
apply Prod_morph; [|simpl plus; reflexivity].
do 2 rewrite subst0_lift. rewrite lift_rec_acc; [simpl; reflexivity|omega].
rewrite H; clear H.
unfold lift. rewrite <- red_lift_prod. apply typ_var; trivial.
Qed.
Lemma Exst_elim : forall e t1 t2 A C,
typ (Nat::e) A prop ->
typ e C prop ->
typ e t1 (Exst A) ->
typ (A::Nat::e) t2 (lift 2 C) ->
exists t, typ e t C.
intros e t1 t2 A C HA HC Ht1 Ht2.
exists (App (App t1 C) (Abs Nat (Abs A t2))).
red; intros i j Hok.
generalize HC; intros HSC.
apply red_typ with (1:=Hok) in HSC; [destruct HSC as (HSC, _)|discriminate].
revert i j Hok; fold (typ e (App (App t1 C) (Abs Nat (Abs A t2))) C).
apply typ_abs in Ht2; [|right|destruct C; [discriminate|]]; trivial.
apply typ_abs in Ht2; [|left; apply typ_N|discriminate].
assert (eq_term C (subst (Abs Nat (Abs A t2)) (lift 1 C))).
unfold subst; rewrite subst_lift_lt; [rewrite lift0; reflexivity|omega].
rewrite H at 2; clear H.
apply typ_app with (V:=(Prod Nat (Prod A (lift 2 C))));
[|unfold Exst in Ht1|discriminate|destruct C; [discriminate|]]; trivial.
assert (eq_term (Prod (Prod Nat (Prod A (lift 2 C))) (lift 1 C))
(subst C (Prod (Prod Nat (Prod (subst (Ref 0) (lift_rec 2 1 A)) ((Ref 2)))) (Ref 1)))).
unfold subst. do 3 rewrite red_sigma_prod.
rewrite (subst0_lift A 1). do 2 (rewrite red_sigma_var_eq; [|trivial]).
apply Prod_morph; [|reflexivity].
apply Prod_morph; [simpl; split; red; reflexivity|].
apply Prod_morph; [|reflexivity].
apply eq_term_intro; [| |destruct A; simpl; trivial]; intros.
rewrite int_subst_rec_eq. rewrite int_lift_rec_eq.
apply int_morph; [reflexivity|do 2 red; intros].
destruct a; unfold V.lams, V.shift; simpl; intros;
[|replace (a-0) with a by omega]; reflexivity.
rewrite tm_subst_rec_eq. rewrite tm_lift_rec_eq.
apply tm_morph; [reflexivity|do 2 red; intros].
destruct a; unfold I.lams, I.shift; simpl; intros;
[|replace (a-0) with a by omega]; reflexivity.
rewrite H; clear H.
apply typ_app with (V:=prop); [| |discriminate|discriminate]; trivial.
Qed.
End FallAndExst.
Section TheorySig.
Lemma typ_S1 : forall e n,
typ e n Nat ->
typ e (App Succ n) Nat.
intros e n Hn.
setoid_replace Nat with (subst n Nat) using relation eq_term;
[|simpl; split; red; reflexivity].
apply typ_app with (V:=Nat); [trivial|apply typ_S|discriminate|discriminate].
Qed.
Lemma int_S : forall n i, int n i ∈ N ->
int (App Succ n) i == succ (int n i).
intros; simpl.
rewrite beta_eq; [reflexivity| |red;rewrite El_def,eqNbot; trivial].
do 2 red; intros; rewrite H1; reflexivity.
Qed.
Definition Add := Abs Nat (Abs Nat (NatRec (Ref 1) (Abs Nat Succ) (Ref 0))).
Lemma typ_Add : forall e, typ e Add (Prod Nat (Prod Nat (lift 2 Nat))).
intros.
apply typ_abs; [left; apply typ_N| |discriminate].
apply typ_abs; [left; apply typ_N| |discriminate].
assert (forall n e, typ e n Nat ->
eq_typ e (App (Abs Nat Nat) n) Nat).
intros. setoid_replace Nat with (subst n Nat) using relation eq_term at 3;
[|simpl; split; red; reflexivity].
apply eq_typ_beta; [apply refl|apply refl|trivial|discriminate].
setoid_replace (lift 2 Nat) with Nat using relation eq_term;
[|simpl; split; red; reflexivity].
apply typ_conv with (T := App (Abs Nat Nat) (Ref 0)); [|apply H|discriminate|discriminate].
apply typ_Nrect.
setoid_replace Nat with (lift 1 Nat) using relation eq_term at 3;
[apply typ_var; trivial|simpl; split; red; reflexivity].
apply typ_conv with (T:=Nat); [| |discriminate|discriminate].
setoid_replace Nat with (lift 2 Nat) using relation eq_term at 3;
[apply typ_var; trivial|simpl; split; red; reflexivity].
apply sym. apply H. apply typ_0.
apply typ_abs; [left; apply typ_N| |discriminate].
apply typ_conv with (T:=Prod Nat (lift 1 Nat)); [apply typ_S| |discriminate|discriminate].
apply sym. apply eq_typ_prod; [| |discriminate].
unfold lift. rewrite red_lift_abs.
setoid_replace (lift_rec 1 0 Nat) with Nat using relation eq_term;
[apply H|simpl; split; red; reflexivity].
setoid_replace Nat with (lift 1 Nat) using relation eq_term at 4;
[apply typ_var; trivial|simpl; split; red; reflexivity].
unfold lift at 2 3. rewrite red_lift_abs.
setoid_replace (lift_rec 1 0 Nat) with Nat using relation eq_term;
[apply H; apply typ_S1|simpl; split; red; reflexivity].
setoid_replace Nat with (lift 2 Nat) using relation eq_term at 6;
[apply typ_var; trivial|simpl; split; red; reflexivity].
setoid_replace Nat with (lift 1 Nat) using relation eq_term at 3;
[apply typ_var; trivial|simpl; split; red; reflexivity].
Qed.
Lemma typ_Add2 : forall e m n,
typ e m Nat ->
typ e n Nat ->
typ e (App (App Add m) n) Nat.
intros e m n Hm Hn.
setoid_replace Nat with (subst n Nat) using relation eq_term;
[|simpl; split; red; reflexivity].
apply typ_app with (V:=Nat); [trivial| |discriminate|discriminate].
setoid_replace (Prod Nat Nat) with (subst m (Prod Nat (lift 2 Nat))) using relation eq_term;
[|simpl; split; red; reflexivity].
apply typ_app with (V:=Nat); [trivial|apply typ_Add|discriminate|discriminate].
Qed.
Lemma int_Add : forall n m i mm nn,
int m i ∈ N ->
int n i ∈ N ->
int m i == mm ->
int n i == nn ->
int (App (App Add m) n) i == add mm nn.
intros n m i mm nn Him Hin Hm Hn.
simpl.
rewrite beta_nat_eq; trivial.
rewrite beta_nat_eq; trivial.
revert nn Hn.
elim Hin using N_ind; intros; auto.
rewrite <- H1.
2:rewrite H0; trivial.
apply natrec_morph_gen2; auto with *.
rewrite natrec_0.
rewrite <- Hn, add0; trivial.
rewrite natrec_S; trivial.
rewrite H0 with (nn:=n0). 2:reflexivity.
rewrite <- Hn, addS; trivial.
rewrite beta_nat_eq; auto.
rewrite beta_nat_eq; auto with *.
apply add_typ; trivial.
rewrite <-Hm; trivial.
do 2 red; intros.
apply natrec_morph_gen2; auto with *.
do 2 red; intros.
apply lam_ext; auto with *.
red; intros.
apply ZFwfr.WFR_morph_gen2; trivial.
red; red; intros.
rewrite H0.
reflexivity.
Qed.
End TheorySig.
Section PresburgerAxioms.
(*Axiom1 : ~ S n = 0*)
Definition ax1_aux := Abs Nat (NatRec True_symb (Abs Nat (Abs prop False_symb)) (Ref 0)).
Lemma P_ax1_aux : forall e, typ e ax1_aux (Prod Nat prop).
intros e. unfold ax1_aux.
assert (forall n e, typ e n Nat ->
eq_typ e (App (Abs Nat prop) n) prop).
intros.
setoid_replace prop with (subst n prop) using relation eq_term at 2;
[apply eq_typ_beta; [apply refl|apply refl|trivial|discriminate]
|simpl; split; red; reflexivity].
apply typ_abs; [left; apply typ_N| |discriminate].
apply typ_conv with (T := App (Abs Nat prop) (Ref 0)); [|apply H|discriminate|discriminate].
apply typ_Nrect.
setoid_replace Nat with (lift 1 Nat) using relation eq_term at 2;
[apply typ_var; trivial|simpl; split; red; reflexivity].
apply typ_conv with (T:=prop); [apply True_symb_typ|apply sym; apply H; apply typ_0
|discriminate|discriminate].
apply typ_abs; [left; apply typ_N| |discriminate].
apply typ_conv with (T:=Prod prop prop); [| |discriminate|discriminate].
apply typ_abs; [left; apply typ_prop|apply False_symb_typ|discriminate].
apply sym; apply eq_typ_prod; [| |discriminate].
unfold lift; rewrite red_lift_abs.
setoid_replace (lift_rec 1 0 Nat) with Nat using relation eq_term;
[|simpl; split; red; reflexivity].
setoid_replace (lift_rec 1 1 prop) with prop using relation eq_term;
[apply H|simpl; split; red; reflexivity].
setoid_replace Nat with (lift 1 Nat) using relation eq_term at 3;
[apply typ_var; trivial|simpl; split; red; reflexivity].
unfold lift at 2; rewrite red_lift_abs.
setoid_replace (lift_rec 2 0 Nat) with Nat using relation eq_term;
[|simpl; split; red; reflexivity].
setoid_replace (lift_rec 2 1 prop) with prop using relation eq_term;
[apply H; apply typ_S1|simpl; split; red; reflexivity].
setoid_replace Nat with (lift 2 Nat) using relation eq_term at 4;
[apply typ_var; trivial|simpl; split; red; reflexivity].
setoid_replace Nat with (lift 1 Nat) using relation eq_term at 2;
[apply typ_var; trivial|simpl; split; red; reflexivity].
Qed.
Lemma ax1_aux_0 : forall e, eq_typ e True_symb (App ax1_aux Zero).
red; intros e i j Hok; simpl.
rewrite beta_nat_eq; [rewrite natrec_0; auto;try reflexivity| |apply zero_typ].
do 2 red; intros; apply natrec_morph_gen2; trivial.
Qed.
Lemma ax1_aux_S : forall e n, typ e n Nat ->
eq_typ e False_symb (App ax1_aux (App Succ n)).
red; intros e n Hn i j Hok.
apply red_typ with (1:=Hok) in Hn; [destruct Hn as (_, (Hn, _))|discriminate].
unfold inX in Hn; rewrite ElNat_eq in Hn.
generalize (True_symb_typ e); intros HT.
apply red_typ with (1:=Hok) in HT; [destruct HT as (_, (HT, _))|discriminate].
unfold inX in HT; simpl int in HT.