Beta-reduction definition and properties.

Require Import base ut_term.
Require Import Peano_dec.
Require Import Compare_dec.
Require Import Lt Plus.

Module Type ut_red_mod (X:term_sig) (TM : ut_term_mod X).
Import TM.

Usual Beta-reduction:

• one step
• multi step (reflexive, transitive closure)
• converesion (reflexive, symetric, transitive closure)
Reserved Notation " A → B " (at level 80).

Inductive Beta : Term -> Term -> Prop :=
| Beta_head : forall A M N , (λ[A], M N M [← N]
| Beta_red1 : forall M M' N , M M' -> M · N M'· N
| Beta_red2 : forall M N N', N N' -> M · N M · N'
| Beta_lam : forall A M M', M M' -> λ[A],M λ[A ],M'
| Beta_lam2 : forall A A' M , A A' -> λ[A],M λ[A'],M
| Beta_pi : forall A B B', B B' -> Π(A),B Π(A ),B'
| Beta_pi2 : forall A A' B , A A' -> Π(A),B Π(A'),B
where "M → N" := (Beta M N) : UT_scope.

Reserved Notation " A →→ B " (at level 80).

Inductive Betas : Term -> Term -> Prop :=
| Betas_refl : forall M , M →→ M
| Betas_Beta : forall M N , M N -> M →→ N
| Betas_trans : forall M N P, M →→ N -> N →→ P -> M →→ P
where " A →→ B " := (Betas A B) : UT_scope.

Reserved Notation " A ≡ B " (at level 80).

Inductive Betac : Term -> Term -> Prop :=
| Betac_Betas : forall M N , M →→ N -> M N
| Betac_sym : forall M N , M N -> N M
| Betac_trans : forall M N P, M N -> N P -> M P
where " A ≡ B " := (Betac A B) : UT_scope.

Hint Constructors Beta.
Hint Constructors Betas.
Hint Constructors Betac.

Facts about Betas and Betac: Congruence.
Lemma Betac_refl : forall M, M M.
intros; constructor; constructor.
Qed.

Hint Resolve Betac_refl.

Lemma Betas_App : forall M M' N N',M →→ M' -> N →→ N' -> M · N →→ M' · N'.
assert (forall a b c, b →→ c -> a·b →→ a·c).
induction 1; eauto.
assert (forall a b c, b→→c -> b· a →→ c· a).
induction 1; eauto.
intros; eauto.
Qed.

Hint Resolve Betas_App.

Lemma Betac_App : forall M M' N N' , M M' -> N N' -> M · N M' · N'.
assert (forall a b c, b c -> a· b a· c).
induction 1; eauto.
assert (forall a b c , b c -> b·a c·a).
induction 1; eauto.
eauto.
Qed.

Hint Resolve Betac_App.

Lemma Betas_La : forall A A' M M', A →→ A' -> M →→ M' -> λ[A],M →→ λ[A'],M'.
assert (forall a b c , a →→ b -> λ[c], a →→ λ[c], b).
induction 1; eauto.
assert (forall a b c, a →→ b -> λ[a], c →→ λ[b], c).
induction 1; eauto.
eauto.
Qed.

Hint Resolve Betas_La.

Lemma Betac_La: forall A A' M M', A A' -> M M' -> λ[A],M λ[A'],M'.
assert (forall a b c, a b -> λ[c], a λ[c], b).
induction 1; eauto.
assert (forall a b c, a b -> λ[a], c λ[b], c).
induction 1; eauto.
eauto.
Qed.

Hint Resolve Betac_La.

Lemma Betas_Pi : forall A A' B B', A →→ A' -> B →→ B' -> Π(A),B →→ Π(A'),B'.
assert (forall a b c , a →→ b -> Π (c), a →→ Π(c), b).
induction 1; eauto.
assert (forall a b c, a →→ b -> Π(a), c →→ Π(b), c).
induction 1; eauto.
eauto.
Qed.

Hint Resolve Betas_Pi.

Lemma Betac_Pi : forall A A' B B', A A' -> B B' -> Π(A),B Π(A'),B'.
assert (forall a b c , a b -> Π(c), a Π(c), b).
induction 1; eauto.
assert (forall a b c, a b -> Π(a), c Π(b), c).
induction 1; eauto.
eauto.
Qed.

Hint Resolve Betac_Pi.

Lemma Beta_beta : forall M N P n, M N -> M[nP] N[nP] .
intros.
generalize n.
induction H; intros; simpl; intuition.
rewrite (subst_travers).
replace (n0+1) with (S n0).
constructor.
rewrite plus_comm. trivial.
Qed.

Some "inversion" lemmas :
• a variable/sort cannot reduce at all
• a pi/lam can only reduce to another pi/lam
• ...
Lemma Betas_V : forall x N, #x →→ N -> N = #x.
intros. remember #x as X; induction H; trivial.
subst; inversion H.
transitivity N. apply IHBetas2. rewrite <- HeqX; intuition. intuition.
Qed.

Lemma Betas_S : forall s N, !s →→ N -> N = !s.
intros. remember !s as S; induction H; trivial.
subst; inversion H.
transitivity N. apply IHBetas2. rewrite <- HeqS; intuition. intuition.
Qed.

Lemma Betas_Pi_inv : forall A B N, Π(A), B →→ N ->
exists C, exists D, N = Π(C), D /\ A →→ C /\ B →→ D.
intros.
remember (Π(A), B) as P. revert A B HeqP; induction H; intros; subst; eauto.
inversion H; subst; clear H.
exists A; exists B'; intuition.
exists A'; exists B; intuition.
destruct (IHBetas1 A B) as (C' & D' & ?); intuition.
destruct (IHBetas2 C' D') as (C'' & D'' &?); intuition.
exists C''; exists D''; eauto.
Qed.

Lift properties.
Lemma Beta_lift: forall M N n m, M N -> M n # m N n # m .
intros.
generalize n m; clear n m.
induction H; intros; simpl; eauto.
change m with (0+m).
rewrite substP1.
constructor.
Qed.

Lemma Betas_lift : forall M N n m , M →→ N -> M n # m →→ N n # m .
intros.
induction H.
constructor.
constructor; apply Beta_lift; intuition.
apply Betas_trans with (N:= N n # m ); intuition.
Qed.

Lemma Betac_lift : forall M N n m, M N -> M n # m N n # m .
intros.
induction H.
constructor.
apply Betas_lift; trivial.
apply Betac_sym; trivial.
apply Betac_trans with (N n # m); trivial.
Qed.

Hint Resolve Beta_lift Betas_lift Betac_lift.

Subst properties.
Lemma Betas_subst : forall M P P' n, P →→ P' -> M [nP] →→ M[n P'].
induction M; intros; simpl; eauto.
destruct (lt_eq_lt_dec v n); intuition.
Qed.

Hint Resolve Betas_subst.

Lemma Betas_subst2 : forall M N P n, M →→ N -> M [n P] →→ N [n P].
induction 1; eauto.
constructor. apply Beta_beta; intuition.
Qed.

Hint Resolve Betas_subst2.

Lemma Betac_subst : forall M P P' n, P P' -> M[nP] M [nP'].
induction M; simpl; intros; intuition.
destruct (lt_eq_lt_dec v n); intuition.
Qed.

Lemma Betac_subst2 : forall M N P n,
M N -> M[nP] N[n P] .
induction 1; eauto.
Qed.

Hint Resolve Betac_subst Betac_subst2.

Parallel Reduction

We use the parallel reduction to prove that Beta is confluent.
Reserved Notation "M →' N" (at level 80).

Beta parallel definition.
Inductive Betap : Term -> Term -> Prop :=
| Betap_sort : forall s , !s →' !s
| Betap_var : forall x , #x →' #x
| Betap_lam : forall A A' M M' , A →' A' -> M →' M' -> λ[A],M →' λ[A'],M'
| Betap_pi : forall A A' B B' , A →' A' -> B →' B' -> Π(A),B →' Π(A'),B'
| Betap_app : forall M M' N N' , M →' M' -> N →' N' -> M · N →' M' · N'
| Betap_head : forall A M M' N N', M →' M' -> N →' N' -> (λ[A],M N →' M'[← N']
where "M →' N" := (Betap M N) : UT_scope.

Notation "m →' n" := (Betap m n) (at level 80) : UT_scope.

Hint Constructors Betap.

Lemma Betap_refl : forall M, M →' M.
induction M; eauto.
Qed.

Hint Resolve Betap_refl.

Lift and Subst property of Betap.
Lemma Betap_lift: forall M N n m, M →' N -> M n # m →' N n # m .
intros.
revert n m.
induction H; simpl; intuition.
change m with (0+m).
rewrite substP1.
constructor; intuition.
Qed.

Hint Resolve Betap_lift.

Lemma Betap_subst:forall M M' N N' n, M →' M' -> N →' N' -> M[nN] →' M'[nN'].
intros. revert n.
induction H; simpl; intuition.
destruct lt_eq_lt_dec; intuition.
rewrite subst_travers. replace (S n) with (n+1) by (rewrite plus_comm; trivial). constructor; intuition.
Qed.

Hint Resolve Betap_subst.

Betap has the diamond property.
Lemma Betap_diamond : forall M N P,
M →' N -> M →' P -> exists Q, N →' Q /\ P →' Q.
intros.
revert P H0.
induction H; intros.
exists P; intuition.
exists P; intuition.
inversion H1; subst; clear H1.
destruct (IHBetap1 A'0 H4) as (A'' & ? & ?).
destruct (IHBetap2 M'0 H6) as (M'' & ?& ?).
exists (λ[A''],M''); intuition.
inversion H1; subst; clear H1.
destruct (IHBetap1 A'0 H4) as (A'' & ? & ?).
destruct (IHBetap2 B'0 H6) as (B'' & ?& ?).
exists (Π(A''), B''); intuition.
inversion H1; subst; clear H1.
destruct (IHBetap1 M'0 H4) as (M'' & ?& ?).
destruct (IHBetap2 N'0 H6) as (N'' & ?& ?).
exists (M'' · N''); intuition.
inversion H; subst; clear H.
destruct (IHBetap2 N'0 H6) as (N'' & ?& ?).
destruct (IHBetap1 (λ[A],M'0)) as (L & ?& ?); intuition.
inversion H2; subst; clear H2; inversion H5; subst; clear H5.
exists ( M' [ N'']); intuition.
inversion H1; subst; clear H1.
destruct (IHBetap2 N'0 H6) as (N'' & ?& ?).
inversion H4; subst; clear H4.
destruct (IHBetap1 M'1 H9) as (M'' & ?& ?).
exists (M''[← N'']); intuition.
destruct (IHBetap2 N'0 H7) as (N'' & ?& ?).
destruct (IHBetap1 M'0 H6) as (M'' & ?& ?).
exists (M''[← N'']); intuition.
Qed.

Reflexive Transitive closure of Betap.
Reserved Notation " x →→' y " (at level 80).

Inductive Betaps : Term -> Term -> Prop :=
| Betaps_refl : forall M , M →→' M
| Betaps_trans : forall M N P, M →' N -> N →→' P -> M →→' P
where " x →→' y " := (Betaps x y) : UT_scope.

Hint Constructors Betaps.

Closure properties to relate Betaps and Betas.
Lemma Betas_Betap_closure : forall M N , M →' N -> M →→ N.
induction 1; eauto.
Qed.

Local Hint Resolve Betas_Betap_closure.

Lemma Betas_Betaps_closure : forall M N, M →→' N -> M →→ N.
induction 1; eauto.
Qed.

Lemma Betap_Beta_closure : forall M N, M N -> M →' N.
induction 1; intuition.
Qed.

Local Hint Resolve Betas_Betaps_closure Betap_Beta_closure.

Lemma Betaps_Beta_closure :forall M N, M N -> M →→' N.
eauto.
Qed.

Local Hint Resolve Betaps_Beta_closure.

Lemma Betaps_trans2 : forall M N P, M →→' N -> N →→' P -> M →→' P.
intros. revert P H0; induction H; eauto.
Qed.

Local Hint Resolve Betaps_trans2.

Lemma Betaps_Betas_closure : forall M N , M →→ N -> M →→' N.
induction 1; eauto.
Qed.

Local Hint Resolve Betaps_Betas_closure.

Betas inherites its diamond property from Betaps.
Lemma sub_diamond_Betaps :
( forall M N P, M →' N -> M →' P -> exists Q, N →' Q /\ P →' Q )
-> forall M N P, M →' N -> M →→' P -> exists Q, N →→' Q /\ P →' Q .
intros.
revert N H0. induction H1; eauto.
intros.
destruct (H M N N0 H0 H2) as (Q & ?& ?).
destruct (IHBetaps Q H3) as (Q'' & ? & ?).
exists Q''; eauto.
Qed.

Lemma diamond_Betaps :
( forall M N P, M →' N -> M →' P -> exists Q, N →' Q /\ P →' Q) ->
forall M N P, M →→' N -> M →→' P -> exists Q, N →→' Q /\ P →→' Q .
intros. revert P H1.
induction H0; intros; eauto.
destruct (sub_diamond_Betaps H M N P0 H0 H2) as (Q & ? & ?).
destruct (IHBetaps Q H3) as (Q'' & ?& ?).
exists Q'';eauto.
Qed.

Theorem Betas_diamond:
forall M N P, M →→ N -> M →→ P -> exists Q, N →→ Q /\ P →→ Q.
intros.
destruct (diamond_Betaps Betap_diamond M N P) as (Q & ?& ?); intuition.
exists Q; intuition.
Qed.

So Beta is confluent.
Theorem Betac_confl : forall M N, M N -> exists Q, M →→ Q /\ N →→ Q.
induction 1; eauto. destruct IHBetac as (Q & ? &? ); eauto.
destruct IHBetac1 as (Q1 & ? &? ), IHBetac2 as (Q2 & ? & ?).
destruct (Betas_diamond N Q1 Q2) as (Q'' & ?& ?); trivial.
exists Q''; eauto.
Qed.

Some consequences:
• convertible sorts are equals
• converitble vars are equals
• Pi-types are Injective
• Pi and sorts are not convertible
Lemma conv_sort : forall s t, !s !t -> s = t.
intros.
apply Betac_confl in H. destruct H as (z & ?& ?).
apply Betas_S in H.
apply Betas_S in H0.
rewrite H in H0.
injection H0; trivial.
Qed.

Lemma conv_to_sort : forall s T, !s T -> T →→ !s.
intros.
apply Betac_confl in H.
destruct H as (z & ?& ?).
apply Betas_S in H.
subst. trivial.
Qed.

Lemma conv_var : forall x y, #x #y -> x = y.
intros.
apply Betac_confl in H. destruct H as (z & ?& ?).
apply Betas_V in H. apply Betas_V in H0.
rewrite H in H0.
injection H0; trivial.
Qed.

Lemma conv_to_var : forall x T , #x T -> T →→ #x.
intros.
apply Betac_confl in H.
destruct H as (z & ?& ?).
apply Betas_V in H.
subst; trivial.
Qed.

Theorem PiInj : forall A B C D, Π(A), B Π(C), D -> A C /\ B D.
intros.
apply Betac_confl in H. destruct H as (z & ? & ?).
apply Betas_Pi_inv in H.
apply Betas_Pi_inv in H0.
destruct H as (c1 & d1 & ? & ? & ?).
destruct H0 as (c2 & d2 & ? & ?& ?).
rewrite H0 in H; injection H; intros; subst; split; eauto.
Qed.

Lemma Betac_not_Pi_sort : forall A B s, ~ (Π(A), B !s ).
intros; intro. apply Betac_sym in H. apply conv_to_sort in H.
apply Betas_Pi_inv in H as (C & D & ? & ? & ?). discriminate.
Qed.

End ut_red_mod.