(**************************************************************************) (* ___ *) (* ||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/relocation/llpx_sn_llpx_sn.ma". include "basic_2/substitution/fqup.ma". include "basic_2/reduction/llpr_ldrop.ma". (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) (* Main properties on context-sensitive parallel reduction for terms ********) fact cpr_conf_llpr_atom_atom: ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T. /2 width=3 by cpr_atom, ex2_intro/ qed-. fact cpr_conf_llpr_atom_delta: ∀G,L0,i. ( ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 → ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T. #G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01 elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_ lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2 lapply (fqup_lref … G … HLK0) -HLK0 #HLK0 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, cpr_delta, ex2_intro/ qed-. (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *) fact cpr_conf_llpr_delta_delta: ∀G,L0,i. ( ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 → ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX → ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T. #G #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 (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_ lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1 elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_ lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2 lapply (fqup_lref … G … HLK0) -HLK0 #HLK0 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/ qed-. fact cpr_conf_llpr_bind_bind: ∀a,I,G,L0,V0,T0. ( ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T. #a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0 elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0 elim (IH … HV01 … HV02 … H1V0 … H2V0) // elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH /3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/ qed-. fact cpr_conf_llpr_bind_zeta: ∀G,L0,V0,T0. ( ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T. #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0 elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0 elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2 elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/ qed-. fact cpr_conf_llpr_zeta_zeta: ∀G,L0,V0,T0. ( ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T. #G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0 elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0 elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2 elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1 elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2 lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/ qed-. fact cpr_conf_llpr_flat_flat: ∀I,G,L0,V0,T0. ( ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T. #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_flat … HL01) -HL01 #H1V0 #H1T0 elim (llpr_inv_flat … HL02) -HL02 #H2V0 #H2T0 elim (IH … HV01 … HV02 … H1V0 … H2V0) // elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/ qed-. fact cpr_conf_llpr_flat_tau: ∀G,L0,V0,T0. ( ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T. #G #L0 #V0 #T0 #IH #V1 #T1 #HT01 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0 elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0 elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/ qed-. fact cpr_conf_llpr_tau_tau: ∀G,L0,V0,T0. ( ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T. #G #L0 #V0 #T0 #IH #T1 #HT01 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0 elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0 elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/ qed-. fact cpr_conf_llpr_flat_beta: ∀a,G,L0,V0,W0,T0. ( ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T. #a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01 elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0 elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02 elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0 elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2 elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *) /4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/ qed-. (* Basic-1: includes: pr0_cong_upsilon_refl pr0_cong_upsilon_zeta pr0_cong_upsilon_cong pr0_cong_upsilon_delta *) fact cpr_conf_llpr_flat_theta: ∀a,G,L0,V0,W0,T0. ( ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. #a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01 elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0 elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02 elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0 elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 elim (lift_total V 0 1) #U #HVU lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2 elim (cpr_inv_abbr1 … H) -H * [ #W1 #T1 #HW01 #HT01 #H destruct elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/ | #T1 #HT01 #HXT1 #H destruct elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/ ] qed-. fact cpr_conf_llpr_beta_beta: ∀a,G,L0,V0,W0,T0. ( ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T. #a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01 #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01 elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0 elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02 elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0 elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2 elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/ lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ /4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *) qed-. (* Basic_1: includes: pr0_upsilon_upsilon *) fact cpr_conf_llpr_theta_theta: ∀a,G,L0,V0,W0,T0. ( ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 → ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. #a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01 elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0 elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02 elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0 elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 elim (lift_total V 0 1) #U #HVU lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/ lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ /4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *) qed-. theorem cpr_conf_llpr: ∀G. llpx_sn_confluent2 (cpr G) (cpr G). #G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ] [ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct elim (cpr_inv_atom1 … H1) -H1 elim (cpr_inv_atom1 … H2) -H2 [ #H2 #H1 destruct /2 width=1 by cpr_conf_llpr_atom_atom/ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct /3 width=10 by cpr_conf_llpr_atom_delta/ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct /4 width=10 by ex2_commute, cpr_conf_llpr_atom_delta/ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct /3 width=17 by cpr_conf_llpr_delta_delta/ ] | #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct elim (cpr_inv_bind1 … H1) -H1 * [ #V1 #T1 #HV01 #HT01 #H1 | #T1 #HT01 #HXT1 #H11 #H12 ] elim (cpr_inv_bind1 … H2) -H2 * [1,3: #V2 #T2 #HV02 #HT02 #H2 |2,4: #T2 #HT02 #HXT2 #H21 #H22 ] destruct [ /3 width=10 by cpr_conf_llpr_bind_bind/ | /4 width=11 by ex2_commute, cpr_conf_llpr_bind_zeta/ | /3 width=11 by cpr_conf_llpr_bind_zeta/ | /3 width=12 by cpr_conf_llpr_zeta_zeta/ ] | #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct elim (cpr_inv_flat1 … H1) -H1 * [ #V1 #T1 #HV01 #HT01 #H1 | #HX1 #H1 | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13 | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13 ] elim (cpr_inv_flat1 … H2) -H2 * [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2 |2,6,10,14: #HX2 #H2 |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23 |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23 ] destruct [ /3 width=10 by cpr_conf_llpr_flat_flat/ | /4 width=8 by ex2_commute, cpr_conf_llpr_flat_tau/ | /4 width=12 by ex2_commute, cpr_conf_llpr_flat_beta/ | /4 width=14 by ex2_commute, cpr_conf_llpr_flat_theta/ | /3 width=8 by cpr_conf_llpr_flat_tau/ | /3 width=8 by cpr_conf_llpr_tau_tau/ | /3 width=12 by cpr_conf_llpr_flat_beta/ | /3 width=13 by cpr_conf_llpr_beta_beta/ | /3 width=14 by cpr_conf_llpr_flat_theta/ | /3 width=17 by cpr_conf_llpr_theta_theta/ ] ] qed-. (* Basic_1: includes: pr0_confluence pr2_confluence *) theorem cpr_conf: ∀G,L. confluent … (cpr G L). /2 width=6 by cpr_conf_llpr/ qed-. (* Properties on context-sensitive parallel reduction for terms *************) lemma llpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 → ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T. #G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cpr_conf_llpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/ qed-. lemma llpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 → ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T. #G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cpr_conf_llpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/ qed-.