(**************************************************************************) (* ___ *) (* ||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/substitution/lpss_cpss.ma". (* SN PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ***************************) (* Main properties on context-sensitive parallel substitution for terms *****) fact cpss_conf_lpss_atom_atom: ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ▶* T. /2 width=3/ qed-. fact cpss_conf_lpss_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 (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 elim (lpss_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 elim (lpss_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 (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ qed-. fact cpss_conf_lpss_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 (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 elim (lpss_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 elim (lpss_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 (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ qed-. fact cpss_conf_lpss_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 cpss_conf_lpss_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-. theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss. #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 (cpss_inv_atom1 … H1) -H1 elim (cpss_inv_atom1 … H2) -H2 [ #H2 #H1 destruct /2 width=1 by cpss_conf_lpss_atom_atom/ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct /3 width=10 by cpss_conf_lpss_atom_delta/ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct /4 width=10 by ex2_commute, cpss_conf_lpss_atom_delta/ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct /3 width=17 by cpss_conf_lpss_delta_delta/ ] | #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct elim (cpss_inv_bind1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /3 width=10 by cpss_conf_lpss_bind_bind/ | #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct elim (cpss_inv_flat1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /3 width=10 by cpss_conf_lpss_flat_flat/ ] qed-. (* Basic_1: was only: subst1_confluence_eq *) theorem cpss_conf: ∀L. confluent … (cpss L). /2 width=6 by cpss_conf_lpss/ qed-. (* Properties on context-sensitive parallel substitution for terms **********) (* Basic_1: was only: subst1_subst1_back *) lemma lpss_cpss_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 (cpss_conf_lpss … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ qed-. lemma lpss_cpss_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 (cpss_conf_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ qed-. (* Main properties **********************************************************) theorem lpss_conf: confluent … lpss. /3 width=6 by lpx_sn_conf, cpss_conf_lpss/ qed-. theorem lpss_trans: Transitive … lpss. /3 width=5 by lpx_sn_trans, cpss_trans_lpss/ qed-. (* Advanced forward lemmas **************************************************) lemma cpss_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