/
ll_cut.v
1266 lines (1245 loc) · 63.3 KB
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ll_cut.v
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(* ll_cut library for yalla *)
(** * Cut admissibility for [ll] *)
Require Import Arith_base.
Require Import Injective.
Require Import List_more.
Require Import List_Type_more.
Require Import Permutation_Type_more.
Require Import genperm_Type.
Require Import flat_map_Type_more.
Require Import wf_nat_more.
Require Export ll_def.
Section Cut_Elim_Proof.
Context {P : pfrag}.
Hypothesis P_gax_at : forall a, Forall atomic (projT2 (pgax P) a).
Lemma cut_oc_comm : pcut P = false -> forall n A l1 l2, ll P (l1 ++ wn A :: l2) ->
(forall lw (pi0 : ll P (dual A :: map wn lw)), psize pi0 < n -> ll P (l1 ++ map wn lw ++ l2)) ->
forall l3 l4 (pi1 : ll P (l3 ++ oc (dual A) :: l4)), psize pi1 <= n -> ll P (l1 ++ l4 ++ l3 ++ l2).
Proof with myeasy_perm_Type.
intros P_cutfree n A l1 l2 pi2 ; induction n ; intros IH l3 l4 pi1 Hs ;
remember (l3 ++ oc (dual A) :: l4) as l ; destruct_ll pi1 f X l Hl Hr HP a ;
try (exfalso ; simpl in Hs ; clear -Hs ; myeasy ; fail) ; try inversion Heql ; subst.
- destruct l3 ; inversion Heql ; subst.
destruct l3 ; inversion H2 ; subst.
destruct l3 ; inversion H3.
- simpl in Hs.
apply PCperm_Type_vs_elt_inv in HP.
destruct HP as [[l3' l4'] Heq HP] ; simpl in Heq ; simpl in HP ; subst.
assert (PEperm_Type (pperm P) (l1 ++ l4' ++ l3' ++ l2) (l1 ++ l4 ++ l3 ++ l2)) as HP'.
{ apply PEperm_Type_app_head.
rewrite 2 app_assoc ; apply PEperm_Type_app_tail.
symmetry... }
apply PEperm_PCperm_Type in HP'.
apply (ex_r _ (l1 ++ l4' ++ l3' ++ l2))...
simpl in Hs ; refine (IHn _ _ _ Hl _)...
intros ; refine (IH _ pi0 _)...
- symmetry in H0 ; trichot_Type_elt_app_exec H0 ; subst.
+ list_simpl ; rewrite app_assoc.
eapply ex_wn_r...
revert Hl Hs ; list_simpl ; intros Hl Hs.
rewrite (app_assoc l5) ; rewrite (app_assoc _ l0) ; rewrite <- (app_assoc l5).
refine (IHn _ _ _ Hl _)...
intros ; refine (IH _ pi0 _)...
+ destruct H2 as [H2 H3] ; simpl in H2 ; decomp_map_Type H2.
inversion H2.
+ list_simpl ; rewrite 2 app_assoc.
eapply ex_wn_r...
revert Hl Hs ; simpl ; rewrite 2 app_assoc ; intros Hl Hs.
list_simpl ; rewrite (app_assoc l) ; rewrite (app_assoc _ l6).
refine (IHn _ _ _ Hl _)...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0.
- dichot_Type_elt_app_exec H0 ; subst.
+ list_simpl.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
simpl in Hs ; refine (IHn _ _ _ Hr _)...
intros ; refine (IH _ pi0 _)...
+ eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
simpl in Hs ; refine (IHn _ _ _ Hl _)...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0.
destruct l3 ; inversion H2.
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply bot_r...
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
simpl in Hs ; refine (IHn _ _ _ Hl _)...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
dichot_Type_elt_app_exec H2 ; subst.
+ list_simpl.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
rewrite app_comm_cons ; eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ] ; list_simpl.
rewrite 3 app_assoc ; apply tens_r...
list_simpl ; rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hr Hs ; simpl ; rewrite (app_comm_cons _ _ B) ; intros Hr Hs.
refine (IHn _ _ _ Hr _)...
intros ; refine (IH _ pi0 _)...
+ eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply tens_r...
list_simpl ; rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl Hs.
refine (IHn _ _ _ Hl _)...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply parr_r...
rewrite 2 app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite (app_comm_cons _ _ B) ; rewrite (app_comm_cons _ _ A0) ; intros Hl Hs.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply top_r...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply plus_r1...
rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl Hs.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply plus_r2...
rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl Hs.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply with_r...
+ rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl Hs.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
+ rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hr Hs ; simpl ; rewrite (app_comm_cons _ _ B) ; intros Hr Hs.
simpl in Hs ; refine (IHn _ _ _ Hr _) ; simpl...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
+ refine (IH _ _ _)...
+ symmetry in H2 ; decomp_map_Type H2.
inversion H2.
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply de_r...
rewrite app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl Hs.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply wk_r...
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
- destruct l3 ; inversion H0 ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
apply co_r...
rewrite 2 app_comm_cons ; eapply ex_r ; [ | apply PCperm_Type_app_rot ] ; list_simpl.
revert Hl Hs ; simpl ; rewrite 2 (app_comm_cons _ _ (wn A0)) ; intros Hl Hs.
simpl in Hs ; refine (IHn _ _ _ Hl _) ; simpl...
intros ; refine (IH _ pi0 _)...
- rewrite f in P_cutfree ; inversion P_cutfree.
- exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat.
apply Forall_app_inv in Hat ; destruct Hat as [_ Hat] ; inversion Hat.
inversion H2.
Qed.
Lemma substitution_oc : pcut P = false -> forall A,
(forall l0 l1 l2, ll P (dual A :: l0) -> ll P (l1 ++ A :: l2) -> ll P (l1 ++ l0 ++ l2)) ->
forall lw l, ll P (dual A :: map wn lw) -> ll P l -> forall l' L,
l = l' ++ flat_map (cons (wn A)) L -> ll P (l' ++ flat_map (app (map wn lw)) L).
Proof with myeasy_perm_Type.
intros P_cutfree A IHcut lw l pi1 pi2.
induction pi2 ; intros l' L Heq ; subst.
- destruct L ; list_simpl in Heq ; subst.
+ list_simpl ; apply ax_r.
+ exfalso.
destruct l' ; inversion Heq.
destruct l' ; inversion H1.
destruct l' ; inversion H3.
- case_eq (pperm P) ; intros Hperm ; rewrite Hperm in p ; simpl in p ; subst.
+ destruct (perm_Type_app_flat_map _ (map wn lw) _ _ _ p) as [[L' l''] (Hnil' & HeqL' & HPL')] ;
simpl in Hnil' ; simpl in HeqL' ; simpl in HPL' ; subst.
eapply ex_r ; [ | rewrite Hperm ; simpl ; apply HPL' ].
apply IHpi2...
+ destruct (cperm_Type_app_flat_map _ (map wn lw) _ _ _ p) as [[L' l''] (Hnil' & HeqL' & HPL')] ;
simpl in Hnil' ; simpl in HeqL' ; simpl in HPL' ; subst.
eapply ex_r ; [ | rewrite Hperm ; simpl ; apply HPL' ].
apply IHpi2...
- assert (injective wn) as Hinj by (intros x y Hxy ; inversion Hxy ; reflexivity).
destruct (perm_flat_map_cons_flat_map_app _ wn Hinj lw _ _ _ _ _ _ p Heq)
as [(((lw1',lw2'),(l1',l2')),(l'',L')) HH] ; simpl in HH ; destruct HH as (H1 & H2 & H3 & H4).
rewrite <- H4 ; apply (ex_wn_r _ _ lw1')...
rewrite H3 ; apply IHpi2...
- symmetry in Heq.
apply app_eq_nil in Heq.
destruct Heq as [Heq' Heq] ; subst.
destruct L ; inversion Heq.
list_simpl ; apply mix0_r...
- app_vs_app_flat_map_inv Heq.
+ list_simpl ; apply mix2_r...
apply IHpi2_1...
+ rewrite flat_map_app ; list_simpl.
rewrite 3 app_assoc ; apply mix2_r...
* apply IHpi2_1...
* list_simpl.
replace (flat_map (app (map wn lw)) L0 ++ map wn lw ++ l)
with (flat_map (app (map wn lw)) (L0 ++ l :: nil))
by (rewrite flat_map_app ; list_simpl ; reflexivity).
apply IHpi2_2...
+ rewrite flat_map_app ; rewrite app_assoc ; apply mix2_r...
* rewrite <- (app_nil_l _).
apply IHpi2_1...
* apply IHpi2_2...
- destruct L ; list_simpl in Heq ; subst.
+ list_simpl ; apply one_r.
+ exfalso.
destruct l' ; inversion Heq.
destruct l' ; inversion H1.
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
simpl ; apply bot_r.
apply IHpi2...
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
app_vs_app_flat_map_inv H1.
+ list_simpl ; apply tens_r...
rewrite app_comm_cons in IHpi2_1 ; rewrite app_comm_cons ; apply IHpi2_1...
+ rewrite flat_map_app ; list_simpl.
rewrite 3 app_assoc ; apply tens_r...
* rewrite app_comm_cons in IHpi2_1 ; rewrite app_comm_cons ; apply IHpi2_1...
* list_simpl.
replace (flat_map (app (map wn lw)) L0 ++ map wn lw ++ l)
with (flat_map (app (map wn lw)) (L0 ++ l :: nil))
by (rewrite flat_map_app ; list_simpl ; reflexivity).
rewrite app_comm_cons in IHpi2_2 ; rewrite app_comm_cons ; apply IHpi2_2...
+ rewrite flat_map_app ; rewrite app_assoc ; simpl ; apply tens_r...
* rewrite <- (app_nil_l (flat_map _ _)) ; rewrite app_comm_cons.
apply IHpi2_1...
* rewrite app_comm_cons in IHpi2_2 ; rewrite app_comm_cons ; apply IHpi2_2...
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
simpl ; apply parr_r.
rewrite 2 app_comm_cons ; apply IHpi2...
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
apply top_r.
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
simpl ; apply plus_r1.
rewrite app_comm_cons ; apply IHpi2...
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
simpl ; apply plus_r2.
rewrite app_comm_cons ; apply IHpi2...
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
simpl ; apply with_r.
+ rewrite app_comm_cons ; apply IHpi2_1...
+ rewrite app_comm_cons ; apply IHpi2_2...
- destruct l' ; inversion Heq ; [ destruct L ; inversion H0 | ] ; subst.
assert ({ Lw | flat_map (app (map wn lw)) L = map wn Lw }) as [Lw HeqLw].
{ clear Heq pi2 IHpi2 ; revert l' H1 ; clear ; induction L ; intros l' Heq.
- exists nil...
- simpl in Heq ; symmetry in Heq ; decomp_map_Type Heq ; subst.
inversion Heq3 ; subst ; simpl.
rewrite Heq5 in IHL ; list_simpl in IHL.
rewrite app_comm_cons in IHL ; rewrite app_assoc in IHL.
destruct (IHL _ eq_refl) as [Lw Heq'].
exists (lw ++ l4 ++ Lw) ; list_simpl ; rewrite <- Heq'... }
symmetry in H1 ; decomp_map_Type H1 ; subst.
list_simpl ; rewrite HeqLw ; rewrite <- map_app ; apply oc_r.
list_simpl in IHpi2 ; simpl in H3 ; rewrite <- H3 in IHpi2.
list_simpl ; rewrite <- HeqLw ;rewrite app_comm_cons ; apply IHpi2...
- destruct l' ; inversion Heq ; subst ; list_simpl.
+ destruct L ; inversion H0 ; subst.
list_simpl.
assert (pi2' := IHpi2 (A :: l0) L eq_refl) ; simpl in pi2'.
rewrite <- (app_nil_l _) in pi2'.
apply (IHcut _ _ _ pi1 pi2').
+ apply de_r.
rewrite app_comm_cons in IHpi2 ; rewrite app_comm_cons ; apply IHpi2...
- destruct l' ; inversion Heq ; subst ; list_simpl.
+ destruct L ; inversion H0 ; subst.
list_simpl.
apply wk_list_r.
apply IHpi2...
+ apply wk_r.
apply IHpi2...
- destruct l' ; inversion Heq ; subst ; list_simpl.
+ destruct L ; inversion H0 ; subst.
list_simpl.
apply co_list_r.
replace (map wn lw ++ map wn lw ++ l0 ++ flat_map (app (map wn lw)) L)
with (nil ++ flat_map (app (map wn lw)) ((nil :: nil) ++ (l0 :: nil) ++ L))
by (rewrite flat_map_app ; list_simpl ; reflexivity).
apply IHpi2...
+ apply co_r.
rewrite 2 app_comm_cons in IHpi2 ; rewrite 2 app_comm_cons ; apply IHpi2...
- rewrite f in P_cutfree ; inversion P_cutfree.
- destruct L ; list_simpl in Heq ; subst.
+ list_simpl ; apply gax_r.
+ exfalso.
specialize P_gax_at with a ; rewrite Heq in P_gax_at.
apply Forall_app_inv in P_gax_at.
destruct P_gax_at as [_ Hat].
inversion Hat ; inversion H1.
Qed.
Hypothesis P_gax_cut : forall a b x l1 l2 l3 l4,
projT2 (pgax P) a = (l1 ++ dual x :: l2) -> projT2 (pgax P) b = (l3 ++ x :: l4) ->
{ c | projT2 (pgax P) c = l3 ++ l2 ++ l1 ++ l4 }.
Theorem cut_r_gaxat : forall A l1 l2,
ll P (dual A :: l1) -> ll P (A :: l2) -> ll P (l2 ++ l1).
Proof with myeasy_perm_Type.
case_eq (pcut P) ; intros P_cutfree.
{ intros A l1 l2 pi1 pi2 ; eapply cut_r... }
enough (forall c s A l0 l1 l2 (pi1 : ll P (dual A :: l0)) (pi2 : ll P (l1 ++ A :: l2)),
s = psize pi1 + psize pi2 -> fsize A <= c -> ll P (l1 ++ l0 ++ l2)) as IH.
{ intros A l1 l2 pi1 pi2.
rewrite <- (app_nil_l _) in pi2.
apply (ex_r _ (nil ++ l1 ++ l2))...
refine (IH _ _ A _ _ _ pi1 pi2 _ _)... }
induction c as [c IHcut0] using lt_wf_rect.
assert (forall A, fsize A < c -> forall l0 l1 l2,
ll P (dual A :: l0) -> ll P (l1 ++ A :: l2) -> ll P (l1 ++ l0 ++ l2)) as IHcut
by (intros A Hs l0 l1 l2 pi1 pi2 ; refine (IHcut0 _ _ _ _ _ _ _ pi1 pi2 _ _) ; myeasy_perm_Type) ;
clear IHcut0.
induction s as [s IHsize0] using lt_wf_rect.
assert (forall A l0 l1 l2 (pi1 : ll P (dual A :: l0)) (pi2 : ll P (l1 ++ A :: l2)),
psize pi1 + psize pi2 < s -> fsize A <= c -> ll P (l1 ++ l0 ++ l2))
as IHsize by (intros ; eapply IHsize0 ; myeasy_perm_Type) ; clear IHsize0.
intros A l0 l1 l2 pi1 pi2 Heqs Hc.
rewrite_all Heqs ; clear s Heqs.
remember (l1 ++ A :: l2) as l ; destruct_ll pi2 f X l Hl Hr HP a.
- (* ax_r *)
destruct l1 ; inversion Heql ; subst.
+ eapply ex_r...
+ unit_vs_elt_inv H1 ; list_simpl...
- (* ex_r *)
simpl in IHsize.
destruct (PCperm_Type_vs_elt_inv _ _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_head _ l0) in HP'.
eapply ex_r ; [ refine (IHsize _ _ _ _ pi1 Hl _ _) | ]...
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; simpl in HP'.
symmetry in HP' ; etransitivity ; [ | etransitivity ; [ apply HP' | ] ]...
- (* ex_wn_r *)
symmetry in Heql ; trichot_Type_elt_app_exec Heql ; list_simpl ; subst.
+ rewrite 2 app_assoc ; eapply ex_wn_r ; [ | apply HP ] ; rewrite <- 2 app_assoc.
revert Hl IHsize ; list_simpl ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _)...
+ destruct Heql1 as [Heql1 Heql2] ; subst.
simpl in Heql1 ; decomp_map_Type Heql1 ; subst ; simpl in HP ; simpl in pi1 ; simpl.
rewrite app_assoc ; rewrite <- (app_nil_l (map wn l7 ++ l3)).
simpl in IHsize.
destruct (Permutation_Type_vs_elt_inv _ _ _ _ HP) as [[lw1 lw2] Heq] ; simpl in Heq ; subst.
revert Hl IHsize ; list_simpl ; rewrite (app_assoc l) ; intros Hl IHsize.
rewrite app_assoc ; rewrite <- (app_nil_l (map wn l7 ++ l3)).
refine (cut_oc_comm _ (psize pi1) x _ _ _ _ _ _ _ _)...
* list_simpl ; replace (map wn l2 ++ wn x :: map wn l7 ++ l3)
with (map wn (l2 ++ x :: l7) ++ l3) by (list_simpl ; reflexivity).
eapply ex_wn_r...
list_simpl ; rewrite app_assoc...
* intros lw pi0 Hs.
list_simpl.
apply Permutation_Type_app_inv in HP.
list_simpl in HP ; apply (Permutation_Type_app_middle lw) in HP.
rewrite (app_assoc (map wn l2)) ; rewrite (app_assoc _ _ l3) ; rewrite <- (app_assoc (map wn l2)).
rewrite <- 2 map_app.
eapply ex_wn_r...
list_simpl ; rewrite app_assoc.
remember (oc_r _ _ _ pi0) as pi0'.
change (oc (dual x)) with (dual (wn x)) in pi0'.
refine (IHsize _ _ _ _ pi0' Hl _ _) ; subst ; simpl...
+ rewrite <- 2 app_assoc.
eapply ex_wn_r ; [ | apply HP ].
rewrite (app_assoc (map wn lw)) ; rewrite (app_assoc l).
revert Hl IHsize ; simpl ; rewrite (app_assoc (map wn lw) l5 _) ; rewrite (app_assoc l) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _)...
- (* mix0_r *)
destruct l1 ; inversion Heql.
- (* mix2_r *)
dichot_Type_elt_app_exec Heql ; subst.
+ rewrite 2 app_assoc ; apply mix2_r...
rewrite <- app_assoc ; refine (IHsize _ _ _ _ pi1 Hr _ _) ; simpl...
+ list_simpl ; apply mix2_r...
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* one_r *)
unit_vs_elt_inv Heql ; list_simpl...
remember (one_r _) as Hone ; clear HeqHone.
remember (dual one :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql' ;
simpl in IHsize.
+ (* ex_r *)
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; simpl in HP'.
eapply ex_r ; [ | etransitivity ; [ apply PCperm_Type_app_comm | symmetry ; apply HP' ] ].
revert Hone IHsize ; change one with (dual bot) ; intros Hone IHsize.
refine (IHsize _ _ _ _ Hone Hl2 _ _)...
+ (* ex_wn_r *)
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
* assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change bot with (dual one) ; intros Hl2 IHsize.
revert Hone IHsize ; rewrite <- (app_nil_l (one :: _)) ; intros Hone IHsize.
replace l0 with (nil ++ l0 ++ nil) by (list_simpl ; reflexivity).
refine (IHsize _ _ _ _ Hl2 Hone _ _)...
* eapply ex_wn_r ; [ | apply HP ].
revert Hl2 IHsize ; simpl ; change bot with (dual one) ; intros Hl2 IHsize.
revert Hone IHsize ; rewrite <- (app_nil_l (one :: _)) ; intros Hone IHsize.
replace (l ++ map wn lw ++ l2) with (nil ++ (l ++ map wn lw ++ l2) ++ nil) by (list_simpl ; reflexivity).
refine (IHsize _ _ _ _ Hl2 Hone _ _)...
+ (* mix2_r *)
destruct l2 ; inversion H0 ; list_simpl in H0 ; subst.
* revert Hl2 IHsize ; change bot with (dual one) ; intros Hl2 IHsize.
revert Hone IHsize ; rewrite <- (app_nil_l (one :: nil)) ; intros Hone IHsize.
replace l0 with (nil ++ l0 ++ nil) by (list_simpl ; reflexivity).
refine (IHsize _ _ _ _ Hl2 Hone _ _)...
* apply mix2_r...
revert Hr2 IHsize ; change bot with (dual one) ; intros Hr2 IHsize.
revert Hone IHsize ; rewrite <- (app_nil_l (one :: nil)) ; intros Hone IHsize.
replace l2 with (nil ++ l2 ++ nil) by (list_simpl ; reflexivity).
refine (IHsize _ _ _ _ Hr2 Hone _ _)...
+ (* bot_r *)
inversion Heql' ; subst...
+ (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
+ (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
- (* bot_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (bot_r _ _ Hl) as Hbot ; clear HeqHbot.
remember (dual bot :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql' ;
simpl in IHsize.
* (* ex_r *)
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hbot IHsize ; change bot with (dual one) ; intros Hbot IHsize.
refine (IHsize _ _ _ _ Hbot Hl2 _ _)...
* (* ex_wn_r *)
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change one with (dual bot) ; intros Hl2 IHsize.
revert Hbot IHsize ; rewrite <- (app_nil_l (bot :: _)) ; intros Hbot IHsize.
refine (IHsize _ _ _ _ Hl2 Hbot _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change one with (dual bot) ; intros Hl2 IHsize.
revert Hbot IHsize ; rewrite <- (app_nil_l (bot :: _)) ; intros Hbot IHsize.
refine (IHsize _ _ _ _ Hl2 Hbot _ _)...
* (* mix2_r *)
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; change one with (dual bot) ; intros Hl2 IHsize.
revert Hbot IHsize ; rewrite <- (app_nil_l (bot :: _)) ; intros Hbot IHsize.
refine (IHsize _ _ _ _ Hl2 Hbot _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; change one with (dual bot) ; intros Hr2 IHsize.
revert Hbot IHsize ; rewrite <- (app_nil_l (bot :: _)) ; intros Hbot IHsize.
refine (IHsize _ _ _ _ Hr2 Hbot _ _)...
* (* one_r *)
list_simpl...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply bot_r.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* tens_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (tens A0 B) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (tens_r _ _ _ _ _ Hl Hr) as Htens ; clear HeqHtens.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ (l4 ++ l3)) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Htens IHsize ; simpl ;
replace (tens A0 B) with (dual (parr (dual B) (dual A0)))
by (simpl ; rewrite 2 bidual ; reflexivity) ;
intros Htens IHsize.
refine (IHsize _ _ _ _ Htens Hl2 _ _)...
simpl in Hc ; simpl ; rewrite 2 fsize_dual...
* (* ex_wn_r *)
remember (tens_r _ _ _ _ _ Hl Hr) as Htens ; clear HeqHtens.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (parr (dual B) (dual A0)) with (dual (tens A0 B)) ;
intros Hl2 IHsize.
revert Htens IHsize ; rewrite <- (app_nil_l (tens _ _ :: _)) ; intros Htens IHsize.
refine (IHsize _ _ _ _ Hl2 Htens _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change (parr (dual B) (dual A0)) with (dual (tens A0 B)) ;
intros Hl2 IHsize.
revert Htens IHsize ; rewrite <- (app_nil_l (tens _ _ :: _)) ; intros Htens IHsize.
refine (IHsize _ _ _ _ Hl2 Htens _ _)...
* (* mix2_r *)
remember (tens_r _ _ _ _ _ Hl Hr) as Htens ; clear HeqHtens.
destruct l2 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (parr (dual B) (dual A0)) with (dual (tens A0 B)) ;
intros Hl2 IHsize.
revert Htens IHsize ; rewrite <- (app_nil_l (tens _ _ :: _)) ; intros Htens IHsize.
refine (IHsize _ _ _ _ Hl2 Htens _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (parr (dual B) (dual A0)) with (dual (tens A0 B)) ;
intros Hr2 IHsize.
revert Htens IHsize ; rewrite <- (app_nil_l (tens _ _ :: _)) ; intros Htens IHsize.
refine (IHsize _ _ _ _ Hr2 Htens _ _)...
* (* parr_r *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; list_simpl.
rewrite <- (bidual B) in Hr.
refine (IHcut _ _ _ _ _ Hr _)...
-- rewrite fsize_dual...
-- eapply ex_r ; [ | apply PCperm_Type_app_comm ].
list_simpl in Hl ; rewrite <- (bidual A0) in Hl.
change ((dual B :: l3) ++ l0) with ((dual B :: nil) ++ l3 ++ l0).
refine (IHcut _ _ _ _ _ Hl _)...
rewrite fsize_dual...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
dichot_Type_elt_app_exec H1 ; subst.
* rewrite 2 app_assoc ; apply tens_r...
revert Hr IHsize ; simpl ; rewrite (app_comm_cons _ _ B) ; intros Hr IHsize.
rewrite <- app_assoc ; refine (IHsize _ _ _ _ pi1 Hr _ _) ; simpl...
* list_simpl ; apply tens_r...
revert Hl IHsize ; simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* parr_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (parr A0 B) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (parr_r _ _ _ _ Hl) as Hparr ; clear HeqHparr.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hparr IHsize ; simpl ;
replace (parr A0 B) with (dual (tens (dual B) (dual A0)))
by (simpl ; rewrite 2 bidual ; reflexivity) ;
intros Hparr IHsize.
refine (IHsize _ _ _ _ Hparr Hl2 _ _)...
simpl in Hc ; simpl ; rewrite 2 fsize_dual...
* (* ex_wn_r *)
remember (parr_r _ _ _ _ Hl) as Hparr ; clear HeqHparr.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (tens (dual B) (dual A0)) with (dual (parr A0 B)) ;
intros Hl2 IHsize.
revert Hparr IHsize ; rewrite <- (app_nil_l (parr _ _ :: _)) ; intros Hparr IHsize.
refine (IHsize _ _ _ _ Hl2 Hparr _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change (tens (dual B) (dual A0)) with (dual (parr A0 B)) ;
intros Hl2 IHsize.
revert Hparr IHsize ; rewrite <- (app_nil_l (parr _ _ :: _)) ; intros Hparr IHsize.
refine (IHsize _ _ _ _ Hl2 Hparr _ _)...
* (* mix2_r *)
remember (parr_r _ _ _ _ Hl) as Hparr ; clear HeqHparr.
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (tens (dual B) (dual A0)) with (dual (parr A0 B)) ;
intros Hl2 IHsize.
revert Hparr IHsize ; rewrite <- (app_nil_l (parr _ _ :: _)) ; intros Hparr IHsize.
refine (IHsize _ _ _ _ Hl2 Hparr _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (tens (dual B) (dual A0)) with (dual (parr A0 B)) ;
intros Hr2 IHsize.
revert Hparr IHsize ; rewrite <- (app_nil_l (parr _ _ :: _)) ; intros Hparr IHsize.
refine (IHsize _ _ _ _ Hr2 Hparr _ _)...
* (* tens_r *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; list_simpl.
refine (IHcut _ _ _ _ _ Hl2 _)...
rewrite <- (app_nil_l _) ; refine (IHcut _ _ _ _ _ Hr2 _)...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply parr_r.
revert Hl IHsize ; simpl ; rewrite (app_comm_cons l1 _ B) ; rewrite (app_comm_cons _ _ A0) ;
intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* top_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual top :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (top_r _ l2) as Htop ; clear HeqHtop.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Htop IHsize ; simpl ; change top with (dual zero) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hplus Hl2 _ _)...
* (* ex_wn_r *)
remember (top_r _ l2) as Htop ; clear HeqHtop.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change zero with (dual top) ; intros Hl2 IHsize.
revert Htop IHsize ; rewrite <- (app_nil_l (top :: _)) ; intros Htop IHsize.
refine (IHsize _ _ _ _ Hl2 Htop _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change zero with (dual top) ; intros Hl2 IHsize.
revert Htop IHsize ; rewrite <- (app_nil_l (top :: _)) ; intros Htop IHsize.
refine (IHsize _ _ _ _ Hl2 Htop _ _)...
* (* mix2_r *)
remember (top_r _ l2) as Htop ; clear HeqHtop.
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change zero with (dual top) ; intros Hl2 IHsize.
revert Htop IHsize ; rewrite <- (app_nil_l (top :: _)) ; intros Htop IHsize.
refine (IHsize _ _ _ _ Hl2 Htop _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change zero with (dual top) ; intros Hr2 IHsize.
revert Htop IHsize ; rewrite <- (app_nil_l (top :: _)) ; intros Htop IHsize.
refine (IHsize _ _ _ _ Hr2 Htop _ _)...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply top_r.
- (* plus_r1 *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (aplus A0 B) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (plus_r1 _ _ _ _ Hl) as Hplus ; clear HeqHplus.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hplus IHsize ; simpl ;
replace (aplus A0 B) with (dual (awith (dual A0) (dual B)))
by (simpl ; rewrite 2 bidual ; reflexivity) ;
intros Hplus IHsize.
refine (IHsize _ _ _ _ Hplus Hl2 _ _)...
simpl ; rewrite 2 fsize_dual...
* (* ex_wn_r *)
remember (plus_r1 _ _ _ _ Hl) as Hplus ; clear HeqHplus.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (awith (dual A0) (dual B)) with (dual (aplus A0 B)) ;
intros Hl2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hl2 Hplus _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change (awith (dual A0) (dual B)) with (dual (aplus A0 B)) ;
intros Hl2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hl2 Hplus _ _)...
* (* mix2_r *)
remember (plus_r1 _ _ _ _ Hl) as Hplus ; clear HeqHplus.
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (awith (dual A0) (dual B)) with (dual (aplus A0 B)) ;
intros Hl2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hl2 Hplus _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (awith (dual A0) (dual B)) with (dual (aplus A0 B)) ;
intros Hr2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hr2 Hplus _ _)...
* (* with_r *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; refine (IHcut _ _ _ _ _ Hl2 Hl)...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply plus_r1.
revert Hl IHsize ; simpl ; rewrite (app_comm_cons l1 _ A0) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* plus_r2 *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (aplus B A0) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (plus_r2 _ _ _ _ Hl) as Hplus ; clear HeqHplus.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hplus IHsize ; simpl ;
replace (aplus B A0) with (dual (awith (dual B) (dual A0)))
by (simpl ; rewrite 2 bidual ; reflexivity) ;
intros Hplus IHsize.
refine (IHsize _ _ _ _ Hplus Hl2 _ _)...
simpl ; rewrite 2 fsize_dual...
* (* ex_wn_r *)
remember (plus_r2 _ _ _ _ Hl) as Hplus ; clear HeqHplus.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (awith (dual B) (dual A0)) with (dual (aplus B A0)) ;
intros Hl2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hl2 Hplus _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change (awith (dual B) (dual A0)) with (dual (aplus B A0)) ;
intros Hl2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hl2 Hplus _ _)...
* (* mix2_r *)
remember (plus_r2 _ _ _ _ Hl) as Hplus ; clear HeqHplus.
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (awith (dual B) (dual A0)) with (dual (aplus B A0)) ;
intros Hl2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hl2 Hplus _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (awith (dual B) (dual A0)) with (dual (aplus B A0)) ;
intros Hr2 IHsize.
revert Hplus IHsize ; rewrite <- (app_nil_l (aplus _ _ :: _)) ; intros Hplus IHsize.
refine (IHsize _ _ _ _ Hr2 Hplus _ _)...
* (* with_r *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; refine (IHcut _ _ _ _ _ Hr2 Hl)...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply plus_r2.
revert Hl IHsize ; simpl ; rewrite (app_comm_cons l1 _ A0) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* with_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (awith A0 B) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (with_r _ _ _ _ Hl Hr) as Hwith ; clear HeqHwith.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hwith IHsize ; simpl ;
replace (awith A0 B) with (dual (aplus (dual A0) (dual B)))
by (simpl ; rewrite 2 bidual ; reflexivity) ;
intros Hwith IHsize.
refine (IHsize _ _ _ _ Hwith Hl2 _ _)...
simpl ; rewrite 2 fsize_dual...
* (* ex_wn_r *)
remember (with_r _ _ _ _ Hl Hr) as Hwith ; clear HeqHwith.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (aplus (dual A0) (dual B)) with (dual (awith A0 B)) ;
intros Hl2 IHsize.
revert Hwith IHsize ; rewrite <- (app_nil_l (awith _ _ :: _)) ; intros Hwith IHsize.
refine (IHsize _ _ _ _ Hl2 Hwith _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change (aplus (dual A0) (dual B)) with (dual (awith A0 B)) ;
intros Hl2 IHsize.
revert Hwith IHsize ; rewrite <- (app_nil_l (awith _ _ :: _)) ; intros Hwith IHsize.
refine (IHsize _ _ _ _ Hl2 Hwith _ _)...
* (* mix2_r *)
remember (with_r _ _ _ _ Hl Hr) as Hwith ; clear HeqHwith.
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (aplus (dual A0) (dual B)) with (dual (awith A0 B)) ;
intros Hl2 IHsize.
revert Hwith IHsize ; rewrite <- (app_nil_l (awith _ _ :: _)) ; intros Hwith IHsize.
refine (IHsize _ _ _ _ Hl2 Hwith _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (aplus (dual A0) (dual B)) with (dual (awith A0 B)) ;
intros Hr2 IHsize.
revert Hwith IHsize ; rewrite <- (app_nil_l (awith _ _ :: _)) ; intros Hwith IHsize.
refine (IHsize _ _ _ _ Hr2 Hwith _ _)...
* (* plus_r1 *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; refine (IHcut _ _ _ _ _ Hl2 Hl)...
* (* plus_r2 *)
clear IHsize ; subst.
rewrite <- (app_nil_l (B :: _)) in Hr ; simpl in Hc ; refine (IHcut _ _ _ _ _ Hl2 Hr)...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply with_r.
* revert Hl IHsize ; simpl ; rewrite (app_comm_cons l1 _ A0) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
* revert Hr IHsize ; simpl ; rewrite (app_comm_cons l1 _ B) ; intros Hr IHsize.
refine (IHsize _ _ _ _ pi1 Hr _ _) ; simpl...
- (* oc_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (oc A0) :: l0) as l' ; destruct_ll pi1 f X lo Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (oc_r _ _ _ Hl) as Hoc ; clear HeqHoc.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ (map wn l)) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hoc IHsize ; simpl ;
replace (oc A0) with (dual (wn (dual A0))) by (simpl ; rewrite bidual ; reflexivity) ;
intros Hoc IHsize.
refine (IHsize _ _ _ _ Hoc Hl2 _ _)...
simpl ; rewrite fsize_dual...
* (* ex_wn_r *)
remember (oc_r _ _ _ Hl) as Hoc ; clear HeqHoc.
destruct lo ; inversion H0 ; [ destruct lw' ; inversion H0 | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (wn (dual A0)) with (dual (oc A0)) ;
intros Hl2 IHsize.
revert Hoc IHsize ; rewrite <- (app_nil_l (oc _ :: _)) ; intros Hoc IHsize.
refine (IHsize _ _ _ _ Hl2 Hoc _ _)...
-- destruct (Permutation_Type_vs_cons_inv _ _ _ HP) as [[lw1 lw2] Heq] ; simpl in Heq ; subst.
assert (Permutation_Type (lw1 ++ l ++ lw2) (l ++ lw')) as HP'.
{ rewrite <- (app_nil_l (l ++ lw')).
apply Permutation_Type_app_middle.
rewrite <- (app_nil_l lw').
apply (Permutation_Type_app_inv _ _ _ _ (dual A0))... }
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
rewrite app_assoc ; rewrite <- map_app ; rewrite <- (app_nil_l _) ; eapply ex_wn_r ; [ | apply HP' ].
list_simpl.
revert Hl2 IHsize ; list_simpl ;intros Hl2 IHsize.
revert Hoc IHsize ; replace (oc A0) with (dual (wn (dual A0)))
by (list_simpl ; rewrite bidual ; reflexivity) ; intros Hoc IHsize.
refine (IHsize _ _ _ _ Hoc Hl2 _ _)...
simpl ; rewrite fsize_dual...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc lo).
revert Hl2 IHsize ; simpl ; change (wn (dual A0)) with (dual (oc A0)) ;
intros Hl2 IHsize.
revert Hoc IHsize ; rewrite <- (app_nil_l (oc _ :: _)) ; intros Hoc IHsize.
refine (IHsize _ _ _ _ Hl2 Hoc _ _)...
* (* mix2_r *)
remember (oc_r _ _ _ Hl) as Hoc ; clear HeqHoc.
destruct l2 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (wn (dual A0)) with (dual (oc A0)) ;
intros Hl2 IHsize.
revert Hoc IHsize ; rewrite <- (app_nil_l (oc _ :: _)) ; intros Hoc IHsize.
refine (IHsize _ _ _ _ Hl2 Hoc _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (wn (dual A0)) with (dual (oc A0)) ;
intros Hr2 IHsize.
revert Hoc IHsize ; rewrite <- (app_nil_l (oc _ :: _)) ; intros Hoc IHsize.
refine (IHsize _ _ _ _ Hr2 Hoc _ _)...
* (* oc_r *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; refine (IHcut _ _ _ _ _ Hl2 Hl)...
* (* wk_r *)
clear IHsize ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
apply wk_list_r...
* (* co_r *)
clear IHsize ; subst.
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
apply co_list_r.
replace (map wn l ++ map wn l ++ l0)
with (nil ++ flat_map (app (map wn l)) (nil :: nil ++ l0 :: nil))
by (list_simpl ; reflexivity).
rewrite <- (bidual A0) in Hl.
refine (substitution_oc _ (dual A0) _ _ _ _ _ _ _ eq_refl) ; list_simpl...
intros l1 l2 l3 pi1 pi2 ; eapply (IHcut (dual A0))...
rewrite fsize_dual ; simpl...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
symmetry in H1 ; decomp_map_Type H1 ; subst ; simpl in pi1 ; simpl in Hl ; simpl.
rewrite app_comm_cons ; rewrite <- (app_nil_l (map wn l6)).
refine (cut_oc_comm _ (psize pi1) x _ _ _ _ _ _ _ _)...
* list_simpl ; replace (map wn l4 ++ wn x :: map wn l6)
with (map wn (l4 ++ x :: l6)) by (list_simpl ; reflexivity).
apply oc_r...
* intros lw pi0 Hs.
list_simpl ; replace (map wn l4 ++ map wn lw ++ map wn l6)
with (map wn (l4 ++ lw ++ l6)) by (list_simpl ; reflexivity).
apply oc_r...
list_simpl ; rewrite app_comm_cons.
remember (oc_r _ _ _ pi0) as pi0'.
change (oc (dual x)) with (dual (wn x)) in pi0'.
revert Hl IHsize ; list_simpl ; rewrite (app_comm_cons _ _ A0) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi0' Hl _ _) ; subst ; simpl...
- (* de_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (wn A0) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (de_r _ _ _ Hl) as Hde ; clear HeqHde.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hde IHsize ; simpl ;
replace (wn A0) with (dual (oc (dual A0))) by (simpl ; rewrite bidual ; reflexivity) ;
intros Hde IHsize.
refine (IHsize _ _ _ _ Hde Hl2 _ _)...
simpl ; rewrite fsize_dual...
* (* ex_wn_r *)
remember (de_r _ _ _ Hl) as Hde ; clear HeqHde.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (oc (dual A0)) with (dual (wn A0)) ;
intros Hl2 IHsize.
revert Hde IHsize ; rewrite <- (app_nil_l (wn _ :: _)) ; intros Hde IHsize.
refine (IHsize _ _ _ _ Hl2 Hde _ _)...
-- list_simpl ; eapply ex_wn_r ; [ | apply HP ] ; rewrite 2 app_assoc ; rewrite <- (app_assoc l).
revert Hl2 IHsize ; simpl ; change (oc (dual A0)) with (dual (wn A0)) ;
intros Hl2 IHsize.
revert Hde IHsize ; rewrite <- (app_nil_l (wn _ :: _)) ; intros Hde IHsize.
refine (IHsize _ _ _ _ Hl2 Hde _ _)...
* (* mix2_r *)
remember (de_r _ _ _ Hl) as Hde ; clear HeqHde.
destruct l3 ; inversion H0 ; list_simpl in H0 ; subst ; list_simpl.
-- revert Hl2 IHsize ; simpl ; change (oc (dual A0)) with (dual (wn A0)) ;
intros Hl2 IHsize.
revert Hde IHsize ; rewrite <- (app_nil_l (wn _ :: _)) ; intros Hde IHsize.
refine (IHsize _ _ _ _ Hl2 Hde _ _)...
-- eapply ex_r ; [ | apply PCperm_Type_app_rot ].
rewrite app_assoc ; apply mix2_r...
eapply ex_r ; [ | apply PCperm_Type_app_comm ].
revert Hr2 IHsize ; simpl ; change (oc (dual A0)) with (dual (wn A0)) ;
intros Hr2 IHsize.
revert Hde IHsize ; rewrite <- (app_nil_l (wn _ :: _)) ; intros Hde IHsize.
refine (IHsize _ _ _ _ Hr2 Hde _ _)...
* (* oc_r *)
clear IHsize ; subst.
rewrite <- (app_nil_l (A0 :: _)) in Hl ; simpl in Hc ; refine (IHcut _ _ _ _ _ Hl2 Hl)...
* (* cut_r *)
rewrite f in P_cutfree ; inversion P_cutfree.
* (* gax_r *)
exfalso.
assert (Hat := P_gax_at a) ; rewrite H0 in Hat ; inversion Hat.
inversion H2.
+ (* commutative case *)
apply de_r.
revert Hl IHsize ; simpl ; rewrite (app_comm_cons l1 _ A0) ; intros Hl IHsize.
refine (IHsize _ _ _ _ pi1 Hl _ _) ; simpl...
- (* wk_r *)
destruct l1 ; inversion Heql ; subst ; list_simpl.
+ (* main case *)
remember (dual (wn A0) :: l0) as l' ; destruct_ll pi1 f X l Hl2 Hr2 HP a ; try inversion Heql'.
* (* ex_r *)
remember (wk_r _ A0 _ Hl) as Hwk ; clear HeqHwk.
destruct (PCperm_Type_vs_cons_inv _ _ _ _ HP) as [[p1 p2] Heq HP'] ; simpl in Heq ; simpl in HP' ; subst.
apply (PEperm_Type_app_tail _ _ _ l2) in HP'.
apply PEperm_PCperm_Type in HP' ; unfold id in HP' ; list_simpl in HP'.
eapply ex_r ; [ | symmetry ; apply HP' ].
eapply ex_r ; [ | symmetry ; apply PCperm_Type_app_rot ].
revert Hwk IHsize ; simpl ;
replace (wn A0) with (dual (oc (dual A0))) by (simpl ; rewrite bidual ; reflexivity) ;
intros Hwk IHsize.
refine (IHsize _ _ _ _ Hwk Hl2 _ _)...
simpl ; rewrite fsize_dual...
* (* ex_wn_r *)
remember (wk_r _ A0 _ Hl) as Hwk ; clear HeqHwk.
destruct l ; inversion Heql' ; [ destruct lw' ; inversion Heql' | ] ; subst.
-- assert (lw = nil) by (clear - HP ; symmetry in HP ; apply (Permutation_Type_nil HP)) ; subst.
list_simpl in Heql' ; subst ; list_simpl in Hl2.
revert Hl2 IHsize ; simpl ; change (oc (dual A0)) with (dual (wn A0)) ;