(**************************************************************************) (* ___ *) (* ||M|| *) (* ||A|| A project by Andrea Asperti *) (* ||T|| *) (* ||I|| Developers: *) (* ||T|| The HELM team. *) (* ||A|| http://helm.cs.unibo.it *) (* \ / *) (* \ / This file is distributed under the terms of the *) (* v GNU General Public License Version 2 *) (* *) (**************************************************************************) include "basic_2/unfold/lpqs_cpqs.ma". (* SN RESTRICTED PARALLEL COMPUTATION ON 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. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ → ∀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 (fsupp_lref … HLK0) -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. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ → ∀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 (fsupp_lref … HLK0) -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. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ → ∀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. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ → ∀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. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ → ∀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. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ → ∀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. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ → ∀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. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ → ∀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 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*] [ #I0 #HL #HT #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 #HL #HT #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 #HL #HT #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-. (* 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-. (* Main properties **********************************************************) theorem lpqs_conf: confluent … lpqs. /3 width=6 by lpx_sn_conf, cpqs_conf_lpqs/ qed-. theorem lpqs_trans: Transitive … lpqs. /3 width=5 by lpx_sn_trans, cpqs_trans_lpqs/ qed-. (* Advanced forward lemmas **************************************************) lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T → ∃∃L2,T2. L @@ L1 ⊢ ➤* L @@ L2 & L @@ L1 ⊢ T1 ➤* T2 & T = L2 @@ T2. #L1 @(lenv_ind_dx … L1) -L1 [ #L #T1 #T #HT1 @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *) | #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H