(**************************************************************************)
include "basic_2/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/relocation/fsup.ma".
+include "basic_2/substitution/fsupp.ma".
include "basic_2/unfold/lpqs_ldrop.ma".
(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
(* Main properties on context-sensitive rest parallel computation for terms *)
-fact cpqs_conf_lpqs_atom_atom:
- ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T.
-/2 width=3/ qed-.
-
-fact cpqs_conf_lpqs_atom_delta:
- ∀L0,i. (
- ∀L,T.♯{L, T} < ♯{L0, #i} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
-qed-.
-
-fact cpqs_conf_lpqs_delta_delta:
- ∀L0,i. (
- ∀L,T.♯{L, T} < ♯{L0, #i} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
- ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
-lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_bind_bind:
- ∀a,I,L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 →
- ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T.
-#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
-qed-.
-
-fact cpqs_conf_lpqs_bind_zeta:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 →
- ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T.
-#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_zeta_zeta:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T.
-#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
-elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
-elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_flat_flat:
- ∀I,L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 →
- ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T.
-#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
-qed-.
-
-fact cpqs_conf_lpqs_flat_tau:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_tau_tau:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
-qed-.
-
-theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs.
-#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
-[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_atom1 … H1) -H1
- elim (cpqs_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpqs_conf_lpqs_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpqs_conf_lpqs_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpqs_conf_lpqs_delta_delta/
- ]
-| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpqs_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpqs_conf_lpqs_bind_bind/
- | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/
- | /3 width=11 by cpqs_conf_lpqs_bind_zeta/
- | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/
- ]
-| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- ]
- elim (cpqs_inv_flat1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #HX2 #H2
- ] destruct
- [ /3 width=10 by cpqs_conf_lpqs_flat_flat/
- | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/
- | /3 width=8 by cpqs_conf_lpqs_flat_tau/
- | /3 width=7 by cpqs_conf_lpqs_tau_tau/
- ]
-]
-qed-.
-
-theorem cpqs_conf: ∀L. confluent … (cpqs L).
-/2 width=6 by cpqs_conf_lpqs/ qed-.
-
theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs.
-#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
-[ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L0 #T0 #IH #L1 * [|*]
+[ #I #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
elim (cpqs_inv_atom1 … H1) -H1
[ #H destruct
elim (cpqs_inv_atom1 … HT2) -HT2
| * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
elim (lpqs_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
elim (lpqs_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
+ lapply (fsupp_lref … HLK1) /3 width=9/
]
| * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
elim (lpqs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
elim (lpqs_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
elim (cpqs_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
+ lapply (fsupp_lref … HLK1) /3 width=9/
]
-| #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
+| #a #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
elim (cpqs_inv_bind1 … H1) -H1 *
[ #V #T #HV1 #HT1 #H destruct
elim (cpqs_inv_bind1 … H2) -H2 *
elim (lift_total X2 0 1) #Y2 #HXY2
lapply (cpqs_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
]
-| #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
+| #I #V1 #T1 #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
elim (cpqs_inv_flat1 … H1) -H1 *
[ #V #T #HV1 #HT1 #H destruct
elim (cpqs_inv_flat1 … H2) -H2 *
(* Properties on context-sensitive rest. parallel computation for terms *****)
-lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
- ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
-qed-.
-
-lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
- ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
-qed-.
-
lemma lpqs_cpqs_trans: ∀L1,L2. L1 ⊢ ➤* L2 →
∀T1,T2. L2 ⊢ T1 ➤* T2 → L1 ⊢ T1 ➤* T2.
/2 width=5 by cpqs_trans_lpqs/ qed-.
-
-lemma fsup_cpqs_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➤* U2 →
- ∃∃L,U1. L1 ⊢ ➤* L & L ⊢ T1 ➤* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-elim (ldrop_lpqs_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
-lapply (cpqs_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-qed-.
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