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SF_reduction.v
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SF_reduction.v
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(**********************************************************************)
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(**********************************************************************)
(**********************************************************************)
(* Intensional Lambda Calculus *)
(* *)
(* is implemented in Coq by adapting the implementation of *)
(* Lambda Calculus from Project Coq *)
(* 2015 *)
(**********************************************************************)
(**********************************************************************)
(* SF_reduction.v *)
(* *)
(* Barry Jay *)
(* *)
(**********************************************************************)
Require Import Arith.
Require Import Test.
Require Import General.
Require Import LamSF_Terms.
Require Import LamSF_Tactics.
Require Import LamSF_Substitution_term.
Require Import Beta_Reduction.
Require Import LamSF_Confluence.
Require Import Omega.
Definition s_op := Op Sop.
Definition f_op := Op Fop.
Definition k_op := App f_op f_op.
Definition i_op := App (App s_op k_op) k_op.
Definition abs_left := App (App s_op k_op) f_op.
Ltac unfold_op := unfold abs_left, i_op, id, k_op, f_op, s_op.
(* components *)
Fixpoint star M :=
match M with
| Ref 0 => i_op
| Ref (S n) => App k_op (Ref n)
| Op o => App k_op (Op o)
| Abs M1 => Abs (star M1)
| App M1 M2 => App (App s_op (Abs M1)) (Abs M2)
end
.
Lemma rank_star:
forall M, rank (star M) < rank (Abs M).
Proof.
induction M; intros.
(* 4 *)
case n; split_all; omega.
(* 3 *)
split_all; omega.
(* 2 *)
unfold star, rank; fold rank; fold star.
eapply2 times_monotonic2.
unfold abs_rank; omega.
eapply2 rank_positive.
(* 1 *)
unfold star; fold star.
unfold_op; unfold rank; fold rank.
replace(abs_rank * S (rank M1 + rank M2)) with (abs_rank + abs_rank * (rank M1) + abs_rank * (rank M2)).
2: split_all; omega.
assert(3 < abs_rank) by (unfold abs_rank; omega).
omega.
Qed.
Lemma star_monotonic:
forall M N,
star M = star N -> M = N.
Proof.
induction M; split_all.
(* 4 *)
gen_case H n.
(* 5 *)
gen_case H N; try discriminate.
(* 5 *)
gen_case H n0; discriminate.
(* 4 *)
gen_case H N.
gen_case H n1.
discriminate.
(* 3 *)
gen_case H N.
gen_case H n.
discriminate.
(* 2 *)
gen_case H N.
gen_case H n.
discriminate.
inversion H.
assert(M = l). eapply2 IHM.
congruence.
(* 1 *)
gen_case H N.
gen_case H n.
discriminate.
Qed.
Fixpoint right_component (M : lamSF) :=
match M with
| App _ M2 => M2
| Abs M1 => star M1
| _ => M
end.
Definition left_component (U : lamSF) :=
match U with
| App U1 _ => U1
| _ => abs_left
end.
Lemma rank_component_app_l:
forall M N, rank (left_component (App M N)) < rank (App M N).
Proof. split_all; omega. Qed.
Lemma rank_component_app_r:
forall M N, rank (right_component (App M N)) < rank (App M N).
Proof. split_all; omega. Qed.
Lemma rank_component_abs_l:
forall M, rank (left_component (Abs M)) < rank (Abs M).
Proof.
induction M; intros.
split_all; try omega.
split_all; try omega.
assert(rank M >0) by eapply2 rank_positive.
unfold left_component. unfold_op; unfold rank; fold rank.
replace (abs_rank * (abs_rank * rank M)) with (abs_rank * abs_rank * rank M) by ring.
cut (S (S (1 + S (1 + 1)) + S (1 + 1)) < abs_rank * abs_rank).
2: unfold abs_rank; split_all; omega. intro.
replace (rank M) with (1+ (pred (rank M))) by omega.
replace(abs_rank * abs_rank * (1+ (pred (rank M)))) with
(abs_rank * abs_rank + abs_rank * abs_rank * (pred (rank M))) by ring.
assert(0 <= abs_rank * abs_rank * pred (rank M)).
case (abs_rank * abs_rank * pred (rank M)); split_all.
omega.
omega.
(* 1 *)
unfold left_component; unfold_op. unfold rank; fold rank.
assert(S (S (1 + S (1 + 1)) + S (1 + 1)) < abs_rank).
unfold abs_rank; split_all; omega.
split_all; omega.
Qed.
Lemma rank_component_abs_r:
forall M, rank (right_component (Abs M)) < rank (Abs M).
Proof.
unfold right_component. intros. eapply2 rank_star.
Qed.
Lemma lift_rec_preserves_star : forall (M : lamSF) n k,
lift_rec(star M) n k = star (lift_rec M (S n) k).
Proof.
induction M; split_all.
case n; split_all.
rewrite relocate_succ. auto.
Qed.
Lemma lift_rec_preserves_components_l : forall (M : lamSF) n k,
lift_rec(left_component M) n k = left_component(lift_rec M n k).
Proof. induction M; split_all; case b0; case b; split_all. Qed.
Lemma lift_rec_preserves_components_r : forall (M : lamSF) n k,
lift_rec(right_component M) n k = right_component(lift_rec M n k).
Proof. induction M; split_all.
rewrite lift_rec_preserves_star.
auto.
Qed.
Lemma star_preserves_status :
forall M, (status (star M) = pred(status M) \/ status(star M) = 0) .
Proof.
induction M; split_all. case n; split_all. inversion IHM. left; omega. right; omega.
Qed.
(* compounds *)
Inductive compound : lamSF -> Prop :=
| op1_compound : forall M o, compound (App (Op o) M)
| op2_compound : forall M N o, compound (App (App (Op o) M) N)
| abs_op_compound : forall o, compound (Abs (Op o))
| abs_compound_compound : forall M, compound M -> compound (Abs M)
| abs_active_compound: forall M, status M = 1 -> compound (Abs M)
.
Hint Resolve
abs_op_compound op1_compound op2_compound abs_compound_compound abs_active_compound
.
Lemma rank_compound_l : forall M, compound M ->
rank (left_component M) < rank M.
Proof.
split_all. inversion H; subst;
eapply2 rank_component_app_l ||
eapply2 rank_component_abs_l ||
split_all.
Qed.
Lemma rank_compound_r : forall M, compound M ->
rank (right_component M) < rank M.
Proof.
split_all. inversion H; subst;
eapply2 rank_component_app_r ||
eapply2 rank_component_abs_r ||
inv1 compound.
Qed.
Lemma lift_rec_preserves_compound :
forall (M: lamSF), compound M -> forall (n k : nat), compound(lift_rec M n k).
Proof.
intros M c; induction c; split_all.
eapply2 abs_active_compound.
assert(status (lift_rec M (S n) k) = relocate (status M) (S (S n)) k).
eapply2 lift_rec_preserves_status.
rewrite H0. rewrite H. relocate_lt. auto.
Qed.
Hint Resolve lift_rec_preserves_compound.
Lemma subst_rec_preserves_compound:
forall (M: lamSF), compound M -> forall N (k : nat), compound(subst_rec M N k).
Proof.
intros M c; induction c; split_all.
eapply2 abs_active_compound.
assert(status (subst_rec M N (S k)) = status M).
eapply2 subst_rec_preserves_status. omega.
rewrite H0. auto.
Qed.
Hint Resolve subst_rec_preserves_compound.
Lemma compound_not_active : forall M, compound M -> status M = 0.
Proof. intros M c; induction c; split_all; omega. Qed.
Lemma star_compound : forall M, compound (star M).
Proof. induction M; split_all; try (case n); split_all; try (unfold_op; auto; fail). Qed.
Lemma subst_rec_preserves_star_active :
forall (M : lamSF) N k, status M <= (S k) ->
subst_rec(star M) N k = star (subst_rec M N (S k)).
Proof.
induction M; split_all.
gen_case H n.
unfold insert_Ref.
elim(compare k n0); elim(compare (S k) (S n0));split_all; try noway.
elim a; elim a0; split_all; try noway.
elim a; split_all; try noway.
elim a; split_all; try noway.
rewrite IHM. auto. omega.
Qed.
Lemma subst_rec_preserves_star_compound :
forall (M : lamSF) N k, compound M ->
subst_rec(star M) N k = star(subst_rec M N (S k)).
Proof.
induction M; split_all; inversion H; split_all.
rewrite IHM; auto.
rewrite subst_rec_preserves_star_active. auto. omega.
Qed.
Lemma subst_rec_preserves_components_l : forall (M : lamSF) n k, compound M ->
subst_rec(left_component M) n k = left_component(subst_rec M n k).
Proof. induction M; split_all; inv1 compound. Qed.
Lemma subst_rec_preserves_components_active_r :
forall (M : lamSF), status M > 0 -> forall n k, status M <= k ->
subst_rec(right_component M) n k = right_component(subst_rec M n k).
Proof.
induction M; split_all.
(* 2 *)
insert_Ref_out; split_all.
rewrite subst_rec_preserves_star_active.
auto.
omega.
Qed.
Lemma subst_rec_preserves_components_compound_r :
forall (M : lamSF), compound M -> forall n k,
subst_rec(right_component M) n k = right_component(subst_rec M n k).
Proof.
induction M; split_all; inversion H; subst; split_all.
rewrite subst_rec_preserves_star_compound.
auto.
auto.
rewrite subst_rec_preserves_star_active.
auto.
omega.
Qed.
Definition preserves_compound (red:termred) :=
forall M, compound M -> forall N, red M N ->
compound N /\ red (left_component M) (left_component N) /\ red(right_component M) (right_component N).
Lemma preserves_compound_seq :
forall (red1 red2:termred), preserves_compound red1 -> preserves_compound red2 ->
preserves_compound (sequential red1 red2).
Proof.
red; split_all.
inversion H2.
elim(H M H1 N0); split_all.
eapply2 H0.
inversion H2.
elim(H M H1 N0); split_all.
elim(H0 N0 H9 N); split_all.
eapply2 seq_red.
inversion H2.
elim(H M H1 N0); split_all.
elim(H0 N0 H9 N); split_all.
eapply2 seq_red.
Qed.
Lemma preserves_compound_multi_step :
forall (red:termred), preserves_compound red -> preserves_compound (multi_step red).
Proof.
red. intros red p M c N R; induction R; split_all.
eapply2 IHR. eapply2 p.
apply succ_red with (left_component N); auto.
eapply2 p. eapply2 IHR. eapply2 p.
apply succ_red with (right_component N); auto.
eapply2 p. eapply2 IHR. eapply2 p.
Qed.
Hint Resolve preserves_compound_multi_step.
(* Operator reduction *)
Inductive opred1 : termred :=
| ref_opred : forall i, opred1 (Ref i) (Ref i)
| op_opred : forall o, opred1 (Op o) (Op o)
| app_opred :
forall M M' ,
opred1 M M' ->
forall N N' : lamSF, opred1 N N' -> opred1 (App M N) (App M' N')
| abs_opred : forall M M' , opred1 M M' -> opred1 (Abs M) (Abs M')
| s_red: forall (M M' N N' P P' : lamSF),
opred1 M M' -> opred1 N N' -> opred1 P P' ->
opred1
(App (App (App s_op M) N) P)
(App (App M' P') (App N' P'))
| f_op_red : forall M M' N o, opred1 M M' ->
opred1 (App (App (App f_op (Op o)) M) N) M'
| f_compound_red: forall (P P' M N N': lamSF), compound P ->
opred1 P P' -> opred1 N N' ->
opred1 (App (App (App f_op P) M) N)
(App (App N' (left_component P')) (right_component P')) .
Hint Resolve
ref_opred op_opred app_opred abs_opred
s_red f_op_red f_compound_red
.
Definition opred := multi_step opred1.
Lemma reflective_opred1 : reflective opred1.
Proof. red; induction M; split_all. Qed.
Hint Resolve reflective_opred1.
Lemma reflective_opred : reflective opred.
Proof. unfold opred; eapply2 refl_multi_step. Qed.
Hint Resolve reflective_opred.
Lemma preserves_abs_opred : preserves_abs opred.
Proof. eapply2 preserves_abs_multi_step; red; split_all. Qed.
Hint Resolve preserves_abs_opred.
Lemma preserves_app_opred : preserves_app opred.
Proof. eapply2 preserves_app_multi_step. red; split_all. Qed.
Hint Resolve preserves_app_opred.
Lemma preserves_star_opred1_active :
forall M, status M > 0 -> forall N, opred1 M N ->
opred1 (star M) (star N).
Proof.
induction M; split_all.
inversion H0.
case n; split_all.
noway.
inversion H0; split_all; subst; simpl in *; try noway.
eapply2 abs_opred.
eapply2 IHM. omega.
inversion H0; split_all; subst; simpl in *; try noway.
assert(status P = 0) by eapply2 compound_not_active. noway.
Qed.
Lemma preserves_star_opred1_compound :
forall M, compound M -> forall N, opred1 M N ->
opred1 (star M) (star N).
Proof.
induction M; split_all; inv1 compound.
(* 6 *)
subst; inversion H0; inversion H2; subst; split_all.
(* 5 *)
unfold_op; split_all. subst; inversion H0; split_all. inversion H1; split_all.
(* 4 *)
subst; inversion H0. assert(opred1 (star M) (star M')) by eapply2 IHM. split_all.
(* 3 *)
subst; inversion H0. simpl in *. eapply2 abs_opred.
eapply2 preserves_star_opred1_active. omega.
(* 2 *)
subst; inversion H0. inversion H2. split_all.
(* 1 *)
subst; inversion H0. inversion H2. inversion H7; split_all.
Qed.
Lemma preserves_status_opred1:
forall M, status M >0 -> forall N, opred1 M N -> status M = status N .
Proof.
cut(forall p M, p>= rank M -> status M >0 -> forall N, opred1 M N -> status M = status N).
split_all; eapply2 H.
induction p; split_all.
assert(rank M >0) by eapply2 rank_positive. noway.
(* p >0 *)
induction M; intros.
inversion H1; split_all.
simpl in *; noway.
inversion H1; split_all.
(* 2 *)
cut(status M = status M'); split_all; eapply2 IHp; simpl in *; omega.
(* 1 *)
generalize IHM1 H H0 H1; clear IHM1 H H0 H1; case M1; intros.
inversion H1. inversion H4; split_all.
simpl in *; noway.
simpl in H0. noway.
(* 1 *)
generalize IHM1 H H0 H1; clear IHM1 H H0 H1; case l; intros.
inversion H1. inversion H4. inversion H9; split_all.
simpl in *; noway.
simpl in *; noway.
(* 1 *)
generalize IHM1 H H0 H1; clear IHM1 H H0 H1; case l1; intros.
inversion H1. inversion H4. inversion H9. inversion H14; split_all.
generalize IHM1 H H0 H1; clear IHM1 H H0 H1; case o; split_all; try noway.
inversion H1. inversion H4. inversion H9. inversion H14; split_all.
eapply2 IHp. omega.
subst. simpl in *. noway.
assert(status l2 = 0) by eapply2 compound_not_active. noway.
simpl in *; noway.
(* 1 *)
assert(status (App (App (App (App l3 l4) l2) l0) M2) = status (App (App (App l3 l4) l2) l0)) by split_all.
rewrite H2. rewrite H2 in H0.
generalize H0; clear H0; inversion H1; intro.
generalize H7; clear H7; inversion H4; intro.
generalize H12; clear H12; inversion H9; intro.
generalize H17; clear H17; inversion H14; intro.
assert(status (App (App (App (App M'2 N'2) N'1) N'0) N') = status (App (App (App M'2 N'2) N'1) N'0)) by split_all.
rewrite H23.
eapply2 IHp.
simpl in *; omega.
(* 9 *)
simpl in *; noway.
simpl in *; noway.
2: simpl in *; noway.
2: simpl in *; noway.
3: simpl in *; noway.
3: simpl in *; noway.
(* 3 *)
simpl in H23.
assert(status P = 0) by eapply2 compound_not_active. noway.
(* 2 *)
simpl in H19.
assert(status P = 0) by eapply2 compound_not_active. noway.
(* 1 *)
simpl in H15.
assert(status l4 = 0) by eapply2 compound_not_active. noway.
Qed.
Lemma preserves_compound_opred1:
forall M, compound M -> forall N, opred1 M N ->
compound N /\
opred1 (left_component M) (left_component N) /\
opred1(right_component M) (right_component N).
Proof.
induction M; split_all.
(* 12 *)
inv1 compound.
inv1 compound.
inv1 compound.
inversion H.
inversion H.
inversion H.
(* 6 *)
inversion H0; subst.
inversion H; subst.
inversion H2; split_all.
eapply2 abs_compound_compound.
eapply2 IHM.
eapply2 abs_active_compound.
rewrite <- H3. eapply2 sym_eq.
eapply2 preserves_status_opred1. omega.
(* 5 *)
inversion H0; subst; split_all.
(* 4 *)
inversion H0; subst.
inversion H; subst.
inversion H2; split_all.
(* 5 *)
simpl in *.
unfold_op; repeat(eapply2 app_opred).
eapply2 preserves_star_opred1_compound.
(* 4 *)
unfold right_component; unfold_op; fold star; repeat(eapply2 app_opred).
eapply2 preserves_star_opred1_active. omega.
(* 3 *)
inversion H; subst.
inversion H0. inversion H3; subst; split_all.
inversion H0. inversion H3; inversion H8; subst; split_all.
(* 2 *)
inversion H; subst.
inversion H0. inversion H3; subst; split_all.
inversion H0. inversion H3; inversion H8; subst; split_all.
(* 1 *)
inversion H; subst.
inversion H0. inversion H3; subst; split_all.
inversion H0. inversion H3; inversion H8; subst; split_all.
Qed.
Hint Resolve preserves_compound_opred1 .
Lemma preserves_compound_opred: preserves_compound opred.
Proof.
eapply2 preserves_compound_multi_step.
split_all; eapply2 preserves_compound_opred1.
Qed.
Hint Resolve preserves_compound_opred .
Lemma lift_rec_preserves_opred1 : lift_rec_preserves opred1.
Proof. red. induction 1; split_all.
relocate_lt.
rewrite lift_rec_preserves_components_l.
rewrite lift_rec_preserves_components_r.
auto.
Qed.
Hint Resolve lift_rec_preserves_opred1.
Lemma lift_rec_preserves_opred : lift_rec_preserves opred.
Proof. eapply2 lift_rec_preserves_multi_step. Qed.
Hint Resolve lift_rec_preserves_opred.
Lemma subst_rec_preserves_opred1 : subst_rec_preserves opred1.
Proof.
red.
intros M M' R; induction R; split_all.
(* 2 *)
unfold insert_Ref. elim(compare k i); split_all. elim a; split_all.
unfold lift. eapply2 lift_rec_preserves_opred1.
(* 1 *)
assert(compound P') by eapply2 preserves_compound_opred1.
rewrite subst_rec_preserves_components_l; split_all.
rewrite subst_rec_preserves_components_compound_r; split_all.
Qed.
Ltac app_op := unfold_op;
match goal with
| |- opred _ _ => red; app_op
| |- multi_step opred1 (Op _) (Op _) => red; one_step; app_op
| |- multi_step opred1 (Abs _) (Abs _) => eapply2 preserves_abs_opred ; app_op
| |- multi_step opred1 (App _ _) (App _ _) => eapply2 preserves_app_opred ; app_op
| |- multi_step opred1 (left_component _) (left_component _) => eapply2 preserves_compound_opred; app_op
| |- multi_step opred1 (right_component _) (right_component _) => eapply2 preserves_compound_opred; app_op
| |- multi_step opred1 (lift_rec _ _ _) (lift_rec _ _ _) => eapply2 lift_rec_preserves_opred; app_op
| |- opred1 (Abs _) (Abs _) => eapply2 abs_opred ; app_op
| |- opred1 (App _ _) (App _ _) => eapply2 app_opred ; app_op
| |- opred1 (left_component _) (left_component _) => eapply2 preserves_compound_opred1; app_op
| |- opred1 (right_component _) (right_component _) => eapply2 preserves_compound_opred1; app_op
| |- opred1 (lift_rec _ _ _) (lift_rec _ _ _) => eapply2 lift_rec_preserves_opred1; app_op
| H : opred1 _ _ |- compound _ => eapply2 preserves_compound_opred1
| |- opred1 (left_component _) _ => eapply2 preserves_compound_opred1
| |- opred1 (right_component _) _ => eapply2 preserves_compound_opred1
| _ => try red; split_all
end.
Ltac opred_compound :=
fold opred in *;
match goal with
| H : opred (App (App (Op ?o) ?M1) ?M2) ?N |- _ =>
assert(opred (right_component (App (App (Op o) M1) M2))
(right_component N)) by
eapply2 preserves_compound_opred;
assert(opred (left_component (App (App (Op o) M1) M2))
(left_component N)) by
eapply2 preserves_compound_opred; simpl in *; clear H; opred_compound
| H : opred (App (Op ?o) ?M1) ?N |- _ =>
assert(opred (right_component (App (Op o) M1))
(right_component N)) by
eapply2 preserves_compound_opred; clear H; opred_compound
| _ => simpl in *
end;
simpl in *.
(* Diamond Lemmas *)
Lemma diamond_opred1_opred1 : diamond opred1 opred1.
Proof.
red; intros M N OR; induction OR; split_all; eauto.
(* 5 subgoals *)
inversion H; clear H; subst; inv opred1; inv opred1; eauto.
(* 8 subgoals *)
elim(IHOR1 M'0); elim(IHOR2 N'0); split_all. eauto.
(* 7 *)
elim(IHOR1 (App (App s_op M'0) N'0));
elim(IHOR2 P'); split_all. unfold s_op in *; inv opred1.
invsub. exist(App (App N'4 x) (App N'3 x)).
(* 6 *)
inversion H7.
elim (IHOR1 (App (App f_op (Op o)) P)); split_all.
unfold f_op in *; inv opred1.
invsub. exist N'1.
(* 5 *)
unfold f_op in *.
elim (IHOR1 (App (App f_op P') N'1)); elim (IHOR2 N'0); split_all.
inv opred1. invsub.
exist(App (App x (left_component N'4)) (right_component N'4)).
eapply2 f_compound_red.
eapply2 preserves_compound_opred1.
app_op.
(* 4 *)
inversion H; subst. elim(IHOR M'0); split_all. exist (Abs x).
(* 3 *)
unfold s_op in *. inv opred1. invsub.
elim(IHOR1 N'2); elim(IHOR2 N'1); elim(IHOR3 N'0); split_all.
exist(App (App x1 x)(App x0 x)).
elim(IHOR1 M'0); elim(IHOR2 N'0); elim(IHOR3 P'0); split_all.
invsub.
exist(App (App x1 x)(App x0 x)).
(* 2 *)
inversion H. inversion H2. inversion H7. inversion H12. inversion H14.
elim(IHOR N'0); split_all. exist x.
elim(IHOR P); split_all.
inversion H3.
(* 1 *)
gen_inv H H0. inversion H2. inversion H8. inversion H13.
elim(IHOR1 N'2); elim(IHOR2 N'0); split_all.
exist (App (App x (left_component x0)) (right_component x0)).
app_op.
eapply2 f_compound_red.
eapply2 preserves_compound_opred1.
inversion H5.
elim(IHOR1 P'0); elim(IHOR2 N'0); split_all.
exist (App (App x (left_component x0)) (right_component x0)).
app_op. app_op.
Qed.
Hint Resolve diamond_opred1_opred1.
Lemma diamond_opred1_opred : diamond opred1 opred.
Proof. eapply2 diamond_strip. Qed.
Lemma diamond_opred : diamond opred opred.
Proof. eapply2 diamond_tiling. Qed.
Hint Resolve diamond_opred.
Definition bop1 := sequential par_red1 opred.
Definition bop := multi_step bop1.
Lemma reflective_bop1: reflective bop1.
Proof. red; reflect. apply seq_red with M; auto. Qed.
Hint Resolve reflective_bop1.
Lemma preserves_abs_bop : preserves_abs bop.
Proof. eapply2 preserves_abs_multi_step; eapply2 preserves_abs_seq; red; split_all. Qed.
Hint Resolve preserves_abs_bop.
Lemma preserves_app_bop1 : preserves_app bop1.
Proof. eapply2 preserves_app_seq; red; split_all. Qed.
Hint Resolve preserves_app_bop1.
Lemma preserves_app_bop : preserves_app bop.
Proof. eapply2 preserves_app_multi_step. Qed.
Hint Resolve preserves_app_bop.
Lemma lift_rec_preserves_bop1 : lift_rec_preserves bop1.
Proof. eapply2 lift_rec_preserves_seq. Qed.
Hint Resolve lift_rec_preserves_bop1.
Lemma lift_rec_preserves_bop : lift_rec_preserves bop.
Proof. eapply2 lift_rec_preserves_multi_step. Qed.
Hint Resolve lift_rec_preserves_bop.
Lemma subst_rec_preserves_bop1 : subst_rec_preserves bop1.
Proof.
eapply2 subst_rec_preserves_seq.
eapply2 subst_rec_preserves_par_red1.
eapply2 subst_rec_preserves_multi_step.
red; split_all; eapply2 subst_rec_preserves_opred1.
red; split_all; eapply2 subst_rec_preserves_opred1.
Qed.
Hint Resolve subst_rec_preserves_bop1.
Lemma subst_rec_preserves_bop : subst_rec_preserves bop.
Proof. eapply2 subst_rec_preserves_multi_step.
red; split_all; eapply2 subst_rec_preserves_bop1.
red; split_all; eapply2 subst_rec_preserves_bop1.
Qed.
Hint Resolve subst_rec_preserves_bop.
(* compounds *)
Lemma preserves_status_par_red1 :
forall M, status M > 0 -> forall N, par_red1 M N -> status M = status N.
Proof.
cut(forall p M, p >= rank M -> status M > 0 -> forall N, par_red1 M N -> status M = status N).
split_all; eapply2 H.
induction p.
split_all. assert(rank M > 0) by eapply2 rank_positive. noway.
(* p > 0 *)
induction M; intros.
inversion H1; split_all.
inversion H1; split_all.
inversion H1; split_all.
assert(status M = status M') .
eapply2 IHM. simpl in *; omega. simpl in *; omega. auto.
(* 1 *)
generalize IHM1 H H0 ; clear IHM1 H H0; inversion H1; intros; try noway.
simpl in H6. noway.
generalize IHM1 H2 H5 H6; clear IHM1 H2 H5 H6. case M1; intros; try noway.
inversion H2; split_all.
inversion H2; split_all.
simpl in H6. noway.
(* 1 *)
generalize IHM1 H5 H6 ; clear IHM1 H5 H6; inversion H2; intros; try noway.
simpl in H11. noway.
subst. clear H1 H2.
(* 1 *)
generalize IHM1 H7 H10 H11; clear IHM1 H7 H10 H11. case l; intros; try noway.
inversion H7; split_all.
inversion H7; split_all.
simpl in H11. noway.
(* 1 *)
generalize IHM1 H7 H10 H11; clear IHM1 H7 H10 H11. case l1; intros; try noway.
inversion H7. inversion H1; split_all.
inversion H7. inversion H1; split_all.
gen2_case IHM1 H11 o.
eapply2 IHp.
simpl in *; omega.
simpl in H11. noway.
(* 1 *)
generalize IHM1 H10 H11 ; clear IHM1 H10 H11; inversion H7; intros; try noway.
generalize IHM1 H10 H11 ; clear IHM1 H10 H11; inversion H1; intros; try noway.
simpl in H13. noway.
assert( status (App (App (App (App l3 l4) l2) l0) M2) = status (App (App (App l3 l4) l2) l0)).
split_all.
rewrite H14 in H13. rewrite H14.
assert( status (App (App (App (App M'1 N'2) N'1) N'0) N') = status(App (App (App M'1 N'2) N'1) N'0)) by split_all.
rewrite H15.
eapply2 IHp.
simpl in *; omega.
Qed.
Lemma preserves_star_par_red1_active :
forall M N, status M > 0 ->
par_red1 M N -> par_red1 (star M) (star N).
Proof.
induction M; split_all.
(* 4 *)
inversion H0; subst; case n; split_all.
noway.
inversion H0; split_all.
eapply2 abs_par_red.
eapply2 IHM. omega.
gen2_inv IHM1 H H0; subst; try noway.
Qed.
Lemma preserves_star_par_red1_compound :
forall M N, compound M ->
par_red1 M N -> par_red1 (star M) (star N).
Proof.
induction M; split_all.
(* 4 *)
inversion H.
inversion H.
(* 2 *)
inversion H0; subst. simpl.
eapply2 abs_par_red.
gen_inv H2 H.
inversion H1. split_all.
eapply2 preserves_star_par_red1_active. omega.
(* 1 *)
gen2_inv IHM1 H H0; subst. inversion H5.
Qed.
Lemma preserves_compound_par_red1 : preserves_compound par_red1.
Proof.
red. induction M; intros.
(* 4 *)
gen_inv H0 H; try (inversion H4); subst; inv par_red1; unfold_op.
(* 3 *)
inversion H.
(* 2 *)
inversion H0; subst.
inversion H; subst.
(* 4 *)
inversion H2; split_all.
elim(IHM H3 M'); split_all.
unfold_op. repeat (eapply2 app_par_red).
eapply2 preserves_star_par_red1_compound.
(* 2 *)
assert(status M = status M'). eapply2 preserves_status_par_red1. omega.
split_all.
eapply2 abs_active_compound.
omega.
eapply2 preserves_star_par_red1_active. omega.
(* 1 *)
inversion H0; subst; split_all; inversion H; subst; split_all;
inversion H3; split_all; inversion H4; split_all.
Qed.
Hint Resolve preserves_compound_par_red1.
Lemma preserves_compound_par_red : preserves_compound par_red.
Proof. red; eapply2 preserves_compound_multi_step. Qed.
Ltac app_par :=
match goal with
| |- par_red1 (left_component _) (left_component _) => eapply2 preserves_compound_par_red1; app_par
| |- par_red1 (right_component _) (right_component _) => eapply2 preserves_compound_par_red1; app_par
| |- par_red1 (Abs _) (Abs _) => eapply2 abs_par_red ; app_par
| |- par_red1 (App _ _) (App _ _) => eapply2 app_par_red ; app_par
| |- par_red1 (subst_rec _ _ _) (subst_rec _ _ _) => eapply2 subst_rec_preserves_par_red1; app_par
| H : par_red1 _ _ |- compound _ => eapply2 preserves_compound_par_red1
| |- par_red1 (left_component _) _ => eapply2 preserves_compound_par_red1
| |- par_red1 (right_component _) _ => eapply2 preserves_compound_par_red1
| _ => try red; split_all
end.
Lemma preserves_compound_bop1: preserves_compound bop1.
Proof. eapply2 preserves_compound_seq. Qed.
Hint Resolve preserves_compound_bop1 .
Lemma preserves_compound_bop: preserves_compound bop.
Proof. eapply2 preserves_compound_multi_step. Qed.
Hint Resolve preserves_compound_bop .
Ltac app_bop := unfold_op;
match goal with
| |- bop _ _ => red; app_bop
| |- multi_step bop1 (Op _) (Op _) => red; one_step; app_bop
| |- multi_step bop1 (Abs _) (Abs _) => eapply2 preserves_abs_opred ; app_bop
| |- multi_step bop1 (App _ _) (App _ _) => eapply2 preserves_app_bop ; app_bop
| |- multi_step bop1 (left_component _) (left_component _) => eapply2 preserves_compound_opred; app_bop
| |- multi_step bop1 (right_component _) (right_component _) => eapply2 preserves_compound_opred; app_bop
| |- multi_step bop1 (lift_rec _ _ _) (lift_rec _ _ _) => eapply2 lift_rec_preserves_opred; app_bop
| |- bop1 (Abs _) (Abs _) => eapply2 abs_opred ; app_bop
| |- bop1 (App _ _) (App _ _) => eapply2 preserves_app_bop ; app_bop
| |- bop1 (left_component _) (left_component _) => eapply2 preserves_compound_bop1; app_bop
| |- bop1 (right_component _) (right_component _) => eapply2 preserves_compound_bop1; app_bop
| |- bop1 (lift_rec _ _ _) (lift_rec _ _ _) => eapply2 lift_rec_preserves_bop1; app_bop
| H : bop1 _ _ |- compound _ => eapply2 preserves_compound_bop1
| |- bop1 (left_component _) _ => eapply2 preserves_compound_bop1
| |- bop1 (right_component _) _ => eapply2 preserves_compound_bop1
| _ => try red; split_all
end.
Ltac bop_compound :=
fold bop in *;
match goal with
| H : bop (App (App (Op ?o) ?M1) ?M2) ?N |- _ =>
assert(bop (right_component (App (App (Op o) M1) M2))
(right_component N)) by
eapply2 preserves_compound_bop;
assert(bop (left_component (App (App (Op o) M1) M2))
(left_component N)) by
eapply2 preserves_compound_bop; simpl in *; clear H; bop_compound
| H : bop (App (Op ?o) ?M1) ?N |- _ =>
assert(bop (right_component (App (Op o) M1))
(right_component N)) by
eapply2 preserves_compound_bop; clear H; bop_compound
| _ => simpl in *
end;
simpl in *.
(* Diamond Lemmas *)
Lemma par_red1_preserves_components_l :
forall M, compound M -> forall N, par_red1 M N ->
par_red1 (left_component M) (left_component N).
Proof. split_all. gen_inv H H0; inv1 compound. Qed.
Lemma par_red1_preserves_components_r :
forall M, compound M -> forall N, par_red1 M N ->
par_red1 (right_component M) (right_component N).
Proof.
split_all. gen_inv H H0; inv1 compound.
subst; inversion H; split_all.
unfold_op; repeat (eapply2 app_par_red).
eapply2 preserves_star_par_red1_compound.
eapply2 preserves_star_par_red1_active. omega.
Qed.
Lemma diamond_opred1_par_red1: diamond opred1 par_red1.
Proof.
red. intros M N R; induction R; split_all; inv par_red1; eauto.
inv opred1; inv par_red1.
(* 6 *)
elim(IHR1 (Abs M'0)); elim(IHR2 N'0); split_all.
inv par_red1. inv opred1.
exist(subst x M').
unfold subst. eapply2 subst_rec_preserves_opred1.
(* 5 *)
elim(IHR1 M'0); elim(IHR2 N'0); split_all. exist(App x0 x).
(* 4 *)
elim(IHR M'0); split_all. exist(Abs x).
(* 3 *)
unfold s_op in *. inv par_red1.
elim(IHR1 N'2); elim(IHR2 N'1); elim(IHR3 N'0); split_all.
exist(App (App x1 x)(App x0 x)).
(* 2 *)
inversion H10. elim(IHR N'0); split_all. exist x.
(* 1 *)
elim(IHR1 N'2); elim(IHR2 N'0); split_all.
exist(App (App x (left_component x0)) (right_component x0)).
eapply2 app_par_red. eapply2 app_par_red.
assert(compound P') by eapply2 preserves_compound_opred1.
eapply2 par_red1_preserves_components_l.
assert(compound P') by eapply2 preserves_compound_opred1.
eapply2 par_red1_preserves_components_r.
inversion H11.
eapply2 f_compound_red.
eapply2 preserves_compound_par_red1.
Qed.
Lemma diamond_opred_par_red1: diamond opred par_red1.
Proof.
eapply2 diamond_flip.