(* *)
(**************************************************************************)
-include "basic_2/unfold/cpqs.ma".
+include "basic_2/notation/relations/pred_3.ma".
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/reduction/lsubx.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
⇩[0, i] L ≡ K. ⓓV → cpr K V V2 →
⇧[0, i + 1] V2 ≡ W2 → cpr L (#i) W2
| cpr_bind : ∀a,I,L,V1,V2,T1,T2.
- cpr L V1 V2 → cpr (L. ⓑ{I} V1) T1 T2 →
- cpr L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+ cpr L V1 V2 → cpr (L.ⓑ{I}V1) T1 T2 →
+ cpr L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
| cpr_flat : ∀I,L,V1,V2,T1,T2.
cpr L V1 V2 → cpr L T1 T2 →
- cpr L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+ cpr L (ⓕ{I} V1. T1) (ⓕ{I}V2.T2)
| cpr_zeta : ∀L,V,T1,T,T2. cpr (L.ⓓV) T1 T →
- ⇧[0, 1] T2 ≡ T → cpr L (+ⓓV. T1) T2
-| cpr_tau : ∀L,V,T1,T2. cpr L T1 T2 → cpr L (ⓝV. T1) T2
-| cpr_beta : ∀a,L,V1,V2,W,T1,T2.
- cpr L V1 V2 → cpr (L.ⓛW) T1 T2 →
- cpr L (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
+ ⇧[0, 1] T2 ≡ T → cpr L (+ⓓV.T1) T2
+| cpr_tau : ∀L,V,T1,T2. cpr L T1 T2 → cpr L (ⓝV.T1) T2
+| cpr_beta : ∀a,L,V1,V2,W1,W2,T1,T2.
+ cpr L V1 V2 → cpr L W1 W2 → cpr (L.ⓛW1) T1 T2 →
+ cpr L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
| cpr_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
cpr L V1 V → ⇧[0, 1] V ≡ V2 → cpr L W1 W2 → cpr (L.ⓓW1) T1 T2 →
- cpr L (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV2. T2)
+ cpr L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
.
interpretation "context-sensitive parallel reduction (term)"
(* Basic properties *********************************************************)
-lemma cpr_lsubr_trans: lsub_trans … cpr lsubr.
+lemma lsubx_cpr_trans: lsub_trans … cpr lsubx.
#L1 #T1 #T2 #H elim H -L1 -T1 -T2
[ //
| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
+ elim (lsubx_fwd_ldrop2_abbr … HL12 … HLK1) -L1 * /3 width=6/
|3,7: /4 width=1/
|4,6: /3 width=1/
|5,8: /4 width=3/
(* Basic_1: was by definition: pr2_free *)
lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
#T1 #T2 #HT12 #L
-lapply (cpr_lsubr_trans … HT12 L ?) //
+lapply (lsubx_cpr_trans … HT12 L ?) //
qed.
-lemma cpqs_cpr: ∀L,T1,T2. L ⊢ T1 ➤* T2 → L ⊢ T1 ➡ T2.
-#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
-qed.
-
-lemma cpss_cpr: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡ T2.
-/3 width=1/ qed.
-
(* Basic_1: includes by definition: pr0_refl *)
lemma cpr_refl: ∀T,L. L ⊢ T ➡ T.
-/2 width=1/ qed.
+#T elim T -T // * /2 width=1/
+qed.
(* Basic_1: was: pr2_head_1 *)
lemma cpr_pair_sn: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 →
∀T. L ⊢ ②{I}V1.T ➡ ②{I}V2.T.
* /2 width=1/ qed.
-lemma cpr_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
+lemma cpr_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓓV) →
∃∃T2,T. L ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
-#L #K #V #T1 #d #HLK
-elim (cpqs_delift … T1 … HLK) -HLK /3 width=4/
+#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/ ]
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i) // /3 width=6/
+| * [ #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/
+ ]
+]
qed-.
lemma cpr_append: l_appendable_sn … cpr.
@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
qed.
-lemma cpr_ext_bind: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
- ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
-#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I
-lapply (cpr_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/
-qed.
-
(* Basic inversion lemmas ***************************************************)
fact cpr_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #L #V #T1 #T #T2 #_ #_ #J #H destruct
| #L #V #T1 #T2 #_ #J #H destruct
-| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #J #H destruct
+| #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
]
qed-.
qed-.
fact cpr_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
+ ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
∃∃V2,T2. L ⊢ V1 ➡ V2 &
- L. ⓑ{I} V1 ⊢ T1 ➡ T2 &
- U2 = ⓑ{a,I} V2. T2
+ L. ⓑ{I}V1 ⊢ T1 ➡ T2 &
+ U2 = ⓑ{a,I}V2.T2
) ∨
∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
#L #U1 #U2 * -L -U1 -U2
[ #I #L #b #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
-| #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
-| #a #L #V1 #V2 #W #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/
+| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
+| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/
+| #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
+| #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
]
qed-.
-lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➡ U2 → (
+lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
∃∃V2,T2. L ⊢ V1 ➡ V2 &
- L. ⓑ{I} V1 ⊢ T1 ➡ T2 &
- U2 = ⓑ{a,I} V2. T2
+ L. ⓑ{I}V1 ⊢ T1 ➡ T2 &
+ U2 = ⓑ{a,I}V2.T2
) ∨
∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
/2 width=3 by cpr_inv_bind1_aux/ qed-.
(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
-lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➡ U2 → (
+lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1.T1 ➡ U2 → (
∃∃V2,T2. L ⊢ V1 ➡ V2 &
- L. ⓓ V1 ⊢ T1 ➡ T2 &
- U2 = ⓓ{a} V2. T2
+ L. ⓓV1 ⊢ T1 ➡ T2 &
+ U2 = ⓓ{a}V2.T2
) ∨
∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
#a #L #V1 #T1 #U2 #H
qed-.
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➡ U2 →
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓛ V1 ⊢ T1 ➡ T2 &
- U2 = ⓛ{a} V2. T2.
+lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L.ⓛV1 ⊢ T1 ➡ T2 &
+ U2 = ⓛ{a}V2.T2.
#a #L #V1 #T1 #U2 #H
elim (cpr_inv_bind1 … H) -H *
[ /3 width=5/
qed-.
fact cpr_inv_flat1_aux: ∀L,U,U2. L ⊢ U ➡ U2 →
- ∀I,V1,U1. U = ⓕ{I} V1. U1 →
+ ∀I,V1,U1. U = ⓕ{I}V1.U1 →
∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
U2 = ⓕ{I} V2. T2
| (L ⊢ U1 ➡ U2 ∧ I = Cast)
- | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
- U1 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
+ L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
+ U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
| ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
- U1 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV2. T2 & I = Appl.
+ U1 = ⓓ{a}W1.T1 &
+ U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
#L #U #U2 * -L -U -U2
[ #I #L #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
-| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
-| #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
-| #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
-| #a #L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=9/
-| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W1 #U1 #H destruct /3 width=13/
+| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
+| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
+| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/
+| #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
+| #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/
+| #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/
+| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/
]
qed-.
-lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. L ⊢ ⓕ{I} V1. U1 ➡ U2 →
+lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. L ⊢ ⓕ{I}V1.U1 ➡ U2 →
∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
- U2 = ⓕ{I} V2. T2
+ U2 = ⓕ{I}V2.T2
| (L ⊢ U1 ➡ U2 ∧ I = Cast)
- | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
- U1 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
+ | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
+ L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
+ U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
| ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
- U1 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV2. T2 & I = Appl.
+ U1 = ⓓ{a}W1.T1 &
+ U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
/2 width=3 by cpr_inv_flat1_aux/ qed-.
(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
-lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐ V1. U1 ➡ U2 →
+lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐV1.U1 ➡ U2 →
∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
- U2 = ⓐ V2. T2
- | ∃∃a,V2,W,T1,T2. L ⊢ V1 ➡ V2 & L.ⓛW ⊢ T1 ➡ T2 &
- U1 = ⓛ{a}W. T1 & U2 = ⓓ{a}V2. T2
+ U2 = ⓐV2.T2
+ | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
+ L.ⓛW1 ⊢ T1 ➡ T2 &
+ U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
| ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
- U1 = ⓓ{a}W1. T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
+ U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
[ /3 width=5/
| #_ #H destruct
-| /3 width=9/
+| /3 width=11/
| /3 width=13/
]
qed-.
#L #V1 #T1 #U #H #HT1
elim (cpr_inv_appl1 … H) -H *
[ /2 width=5/
-| #a #V2 #W #U1 #U2 #_ #_ #H #_ destruct
+| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
elim (simple_inv_bind … HT1)
| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
elim (simple_inv_bind … HT1)
#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
[ /3 width=5/
| /2 width=1/
-| #a #V2 #W #T1 #T2 #_ #_ #_ #_ #H destruct
+| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
]
qed-.
(* Basic forward lemmas *****************************************************)
-lemma cpr_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀I,W.
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 &
- U2 = ⓛ{a} V2. T2.
-#a #L #V1 #T1 #U2 #H #I #W
-elim (cpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-lapply (cpr_lsubr_trans … HT12 (L.ⓑ{I}W) ?) -HT12 /2 width=1/ /2 width=5/
-qed-.
-
-
-lemma cpr_fwd_ext_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
- U2 = ⓛ{a}V2.T2.
-#a #L #V1 #T1 #U2 #H #b #I #W
-elim (cpr_fwd_abst1 … H I W) -H /3 width=5/
-qed-.
-
lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
T = -ⓑ{I}V2.T2.
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