include "basic_2/notation/relations/pred_4.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
-include "basic_2/substitution/lsubr.ma".
+include "basic_2/static/lsubr.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
inductive cpr: relation4 genv lenv term term ≝
| cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
| cpr_delta: ∀G,L,K,V,V2,W2,i.
- ⇩[0, i] L ≡ K. ⓓV → cpr G K V V2 →
+ ⇩[i] L ≡ K. ⓓV → cpr G K V V2 →
⇧[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
| cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
| cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
⇧[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
-| cpr_tau : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
+| cpr_eps : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
| cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
[ //
| #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -L1 * /3 width=6/
-|3,7: /4 width=1/
-|4,6: /3 width=1/
-|5,8: /4 width=3/
+ elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 *
+ /3 width=6 by cpr_delta/
+|3,7: /4 width=1 by lsubr_bind, cpr_bind, cpr_beta/
+|4,6: /3 width=1 by cpr_flat, cpr_eps/
+|5,8: /4 width=3 by lsubr_bind, cpr_zeta, cpr_theta/
]
qed-.
(* Basic_1: includes by definition: pr0_refl *)
lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
-#G #T elim T -T // * /2 width=1/
+#G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/
qed.
(* Basic_1: was: pr2_head_1 *)
lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
-* /2 width=1/ qed.
+* /2 width=1 by cpr_bind, cpr_flat/ qed.
-lemma cpr_delift: ∀G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓓV) →
+lemma cpr_delift: ∀G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓓV) →
∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
#G #K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK /2 width=4/
- elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
+[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
+ #i #L #d #HLK elim (lt_or_eq_or_gt i d)
+ #Hid [1,3: /3 width=4 by cpr_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
destruct
elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i) // /3 width=6/
+ elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+ [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
]
]
qed-.
-lemma cpr_append: ∀G. l_appendable_sn … (cpr G).
-#G #K #T1 #T2 #H elim H -G -K -T1 -T2 // /2 width=1/ /2 width=3/
-#G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpr_delta … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
-qed.
-
(* Basic inversion lemmas ***************************************************)
fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+ ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
#G #L #T1 #T2 * -G -L -T1 -T2
-[ #I #G #L #J #H destruct /2 width=1/
-| #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
+[ #I #G #L #J #H destruct /2 width=1 by or_introl/
+| #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+ ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
/2 width=3 by cpr_inv_atom1_aux/ qed-.
(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
T2 = #i ∨
- ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+ ∃∃K,V,V2. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
⇧[O, i + 1] V2 ≡ T2.
#G #L #T2 #i #H
-elim (cpr_inv_atom1 … H) -H /2 width=1/
-* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
+elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
qed-.
lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
#G #L #U1 #U2 * -L -U1 -U2
[ #I #G #L #b #J #W1 #U1 #H destruct
| #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/
+| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
-| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/
+| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
) ∨
∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
#a #G #L #V1 #T1 #U2 #H
-elim (cpr_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
+elim (cpr_inv_bind1 … H) -H *
+/3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
qed-.
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
U2 = ⓛ{a}V2.T2.
#a #G #L #V1 #T1 #U2 #H
elim (cpr_inv_bind1 … H) -H *
-[ /3 width=5/
+[ /3 width=5 by ex3_2_intro/
| #T #_ #_ #_ #H destruct
]
qed-.
[ #I #G #L #J #W1 #U1 #H destruct
| #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/
+| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/
| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
-| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/
+| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or4_intro2, ex6_6_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or4_intro3, ex7_7_intro/
]
qed-.
⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
-[ /3 width=5/
+[ /3 width=5 by or3_intro0, ex3_2_intro/
| #_ #H destruct
-| /3 width=11/
-| /3 width=13/
+| /3 width=11 by or3_intro1, ex5_6_intro/
+| /3 width=13 by or3_intro2, ex6_7_intro/
]
qed-.
U2 = ⓝ V2. T2
) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
-[ /3 width=5/
-| /2 width=1/
+[ /3 width=5 by ex3_2_intro, or_introl/
+| /2 width=1 by or_intror/
| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
]
T = -ⓑ{I}V2.T2.
#I #G #L #V1 #T1 #T #H #b
elim (cpr_inv_bind1 … H) -H *
-[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/
| #T2 #_ #_ #H destruct
]
qed-.
-lemma cpr_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpr_inv_bind1 … H) -H *
- [ #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
-
(* Basic_1: removed theorems 11:
pr0_subst0_back pr0_subst0_fwd pr0_subst0
pr2_head_2 pr2_cflat clear_pr2_trans
pr2_gen_ctail pr2_ctail
*)
(* Basic_1: removed local theorems 4:
- pr0_delta_tau pr0_cong_delta
+ pr0_delta_eps pr0_cong_delta
pr2_free_free pr2_free_delta
*)
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