(* *)
(**************************************************************************)
-include "basic_2/rt_computation/cnuw.ma".
-include "basic_2/rt_computation/cprs_tweq.ma".
+include "basic_2/rt_computation/cnuw_simple.ma".
+include "basic_2/rt_computation/cnuw_drops.ma".
+include "basic_2/rt_computation/cprs_teqw.ma".
include "basic_2/rt_computation/lprs_cpms.ma".
(* NORMAL TERMS FOR T-UNUNBOUND WHD RT-TRANSITION ***************************)
(* Advanced inversion lemmas ************************************************)
lemma cnuw_inv_abbr_pos (h) (G) (L):
- â\88\80V,T. â¦\83G,Lâ¦\84 ⊢ ➡𝐍𝐖*[h] +ⓓV.T → ⊥.
+ â\88\80V,T. â\9d¨G,Lâ\9d© ⊢ ➡𝐍𝐖*[h] +ⓓV.T → ⊥.
#h #G #L #V #T1 #H
-elim (cprs_abbr_pos_twneq h G L V T1) #T2 #HT12 #HnT12
+elim (cprs_abbr_pos_tneqw h G L V T1) #T2 #HT12 #HnT12
/3 width=2 by/
qed-.
(* Advanced properties ******************************************************)
-lemma cnuw_abbr_neg (h) (G) (L): â\88\80V,T. â¦\83G,Lâ¦\84 ⊢ ➡𝐍𝐖*[h] -ⓓV.T.
+lemma cnuw_abbr_neg (h) (G) (L): â\88\80V,T. â\9d¨G,Lâ\9d© ⊢ ➡𝐍𝐖*[h] -ⓓV.T.
#h #G #L #V1 #T1 #n #X #H
elim (cpms_inv_abbr_sn_dx … H) -H *
-[ #V2 #T2 #_ #_ #H destruct /1 width=1 by tweq_abbr_neg/
+[ #V2 #T2 #_ #_ #H destruct /1 width=1 by teqw_abbr_neg/
| #X1 #_ #_ #H destruct
]
qed.
-lemma cnuw_abst (h) (p) (G) (L): â\88\80W,T. â¦\83G,Lâ¦\84 â\8a¢ â\9e¡ð\9d\90\8dð\9d\90\96*[h] â\93\9b{p}W.T.
+lemma cnuw_abst (h) (p) (G) (L): â\88\80W,T. â\9d¨G,Lâ\9d© â\8a¢ â\9e¡ð\9d\90\8dð\9d\90\96*[h] â\93\9b[p]W.T.
#h #p #G #L #W1 #T1 #n #X #H
elim (cpms_inv_abst_sn … H) -H #W2 #T2 #_ #_ #H destruct
-/1 width=1 by tweq_abst/
+/1 width=1 by teqw_abst/
qed.
lemma cnuw_cpms_trans (h) (n) (G) (L):
- â\88\80T1. â¦\83G,Lâ¦\84 ⊢ ➡𝐍𝐖*[h] T1 →
- â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ➡𝐍𝐖*[h] T2.
+ â\88\80T1. â\9d¨G,Lâ\9d© ⊢ ➡𝐍𝐖*[h] T1 →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,n] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ ➡𝐍𝐖*[h] T2.
#h #n1 #G #L #T1 #HT1 #T2 #HT12 #n2 #T3 #HT23
-/4 width=5 by cpms_trans, tweq_canc_sn/
+/4 width=5 by cpms_trans, teqw_canc_sn/
+qed-.
+
+lemma cnuw_dec_ex (h) (G) (L):
+ ∀T1. ∨∨ ❨G,L❩ ⊢ ➡𝐍𝐖*[h] T1
+ | ∃∃n,T2. ❨G,L❩ ⊢ T1 ➡*[h,n] T2 & (T1 ≃ T2 → ⊥).
+#h #G #L #T1 elim T1 -T1 *
+[ #s /3 width=5 by cnuw_sort, or_introl/
+| #i elim (drops_F_uni L i)
+ [ /3 width=7 by cnuw_atom_drops, or_introl/
+ | * * [ #I | * #V ] #K #HLK
+ [ /3 width=8 by cnuw_unit_drops, or_introl/
+ | elim (lifts_total V 𝐔❨↑i❩) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpms_delta_drops/ ] #H
+ lapply (teqw_inv_lref_sn … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ | elim (lifts_total V 𝐔❨↑i❩) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpms_ell_drops/ ] #H
+ lapply (teqw_inv_lref_sn … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ ]
+ ]
+| #l /3 width=5 by cnuw_gref, or_introl/
+| #p * [ cases p ] #V1 #T1 #_ #_
+ [ elim (cprs_abbr_pos_tneqw h G L V1 T1) #T2 #HT12 #HnT12
+ /4 width=4 by ex2_2_intro, or_intror/
+ | /3 width=5 by cnuw_abbr_neg, or_introl/
+ | /3 width=5 by cnuw_abst, or_introl/
+ ]
+| * #V1 #T1 #_ #IH
+ [ elim (simple_dec_ex T1) [ #HT1 | * #p * #W1 #U1 #H destruct ]
+ [ elim IH -IH
+ [ /3 width=6 by cnuw_appl_simple, or_introl/
+ | * #n #T2 #HT12 #HnT12 -HT1
+ @or_intror @(ex2_2_intro … n (ⓐV1.T2)) [ /2 width=1 by cpms_appl_dx/ ] #H
+ lapply (teqw_inv_appl_bi … H) -H /2 width=1 by/
+ ]
+ | elim (lifts_total V1 𝐔❨1❩) #X1 #HVX1
+ @or_intror @(ex2_2_intro … (ⓓ[p]W1.ⓐX1.U1)) [1,2: /2 width=3 by cpms_theta/ ] #H
+ elim (teqw_inv_appl_sn … H) -H #X1 #X2 #_ #H destruct
+ | @or_intror @(ex2_2_intro … (ⓓ[p]ⓝW1.V1.U1)) [1,2: /2 width=2 by cpms_beta/ ] #H
+ elim (teqw_inv_appl_sn … H) -H #X1 #X2 #_ #H destruct
+ ]
+ | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpms_eps/ ] #H
+ /2 width=4 by teqw_inv_cast_xy_y/
+ ]
+]
+qed-.
+
+lemma cnuw_dec (h) (G) (L): ∀T. Decidable (❨G,L❩ ⊢ ➡𝐍𝐖*[h] T).
+#h #G #L #T1
+elim (cnuw_dec_ex h G L T1)
+[ /2 width=1 by or_introl/
+| * #n #T2 #HT12 #nT12 /4 width=2 by or_intror/
+]
qed-.