X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Flpr_lpr.ma;h=d11e148d16eae1df7b447c4fc77357b2ec5f5cb7;hb=d4a90dfb8d8a56012928a600ea2f6bd4758b51f6;hp=ab77692ddd871ccc19f8fb8077bf3a394a117c2c;hpb=b9d48c1a4433b3a2b17483bc207b2092f7ac4fd2;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma index ab77692dd..d11e148d1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma @@ -12,9 +12,338 @@ (* *) (**************************************************************************) -include "basic_2/reduction/lpr_cpr.ma". +include "basic_2/grammar/lpx_sn_lpx_sn.ma". +include "basic_2/relocation/fsup.ma". +include "basic_2/reduction/lpr_ldrop.ma". -(* SN PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ******************************) +(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) + +(* Main properties on context-sensitive parallel reduction for terms ********) + +fact cpr_conf_lpr_atom_atom: + ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➡ T & L2 ⊢ ⓪{I} ➡ T. +/2 width=3/ qed-. + +fact cpr_conf_lpr_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 (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct +elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpr_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 (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ +qed-. + +(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *) +fact cpr_conf_lpr_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 (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 +elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpr_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 (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 +lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ +qed-. + +fact cpr_conf_lpr_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 cpr_conf_lpr_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 (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/ +qed-. + +fact cpr_conf_lpr_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 (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1 +elim (cpr_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 cpr_conf_lpr_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 cpr_conf_lpr_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 cpr_conf_lpr_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-. + +fact cpr_conf_lpr_flat_beta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.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 ⊢ ⓛ{a}W0.T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW1)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/ +qed-. + +(* Basic-1: includes: + pr0_cong_upsilon_refl pr0_cong_upsilon_zeta + pr0_cong_upsilon_cong pr0_cong_upsilon_delta +*) +fact cpr_conf_lpr_flat_theta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.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 ⊢ ⓓ{a}W0.T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → + ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H +#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (lift_total V 0 1) #U #HVU +lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ #HU2 +elim (cpr_inv_abbr1 … H) -H * +[ #W1 #T1 #HW01 #HT01 #H destruct + elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ + elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 /4 width=7/ +| #T1 #HT01 #HXT1 #H destruct + elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 + elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /2 width=1/ #Y #HYT #HXY + @(ex2_intro … (ⓐV.Y)) /2 width=1/ /3 width=5/ (**) (* auto /4 width=9/ is too slow *) +] +qed-. + +fact cpr_conf_lpr_beta_beta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.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.ⓛW0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HT01 … HT02 (L1.ⓛW0) … (L2.ⓛW0)) /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 +lapply (cpr_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ +lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/ +qed-. + +(* Basic_1: was: pr0_upsilon_upsilon *) +fact cpr_conf_lpr_theta_theta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.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 → ∀U1. ⇧[O, 1] V1 ≡ U1 → + ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → + ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. +#a #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 (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ +elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 +elim (lift_total V 0 1) #U #HVU +lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ +lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ /4 width=7/ +qed-. + +theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr. +#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] +[ #I0 #Hn #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_lpr_atom_atom/ + | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct + /3 width=10 by cpr_conf_lpr_atom_delta/ + | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct + /4 width=10 by ex2_commute, cpr_conf_lpr_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_lpr_delta_delta/ + ] +| #a #I #V0 #T0 #Hn #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_lpr_bind_bind/ + | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/ + | /3 width=11 by cpr_conf_lpr_bind_zeta/ + | /3 width=12 by cpr_conf_lpr_zeta_zeta/ + ] +| #I #V0 #T0 #Hn #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 #Z1 #T1 #HV01 #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 #Z2 #T2 #HV02 #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_lpr_flat_flat/ + | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/ + | /4 width=11 by ex2_commute, cpr_conf_lpr_flat_beta/ + | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/ + | /3 width=8 by cpr_conf_lpr_flat_tau/ + | /3 width=7 by cpr_conf_lpr_tau_tau/ + | /3 width=11 by cpr_conf_lpr_flat_beta/ + | /3 width=11 by cpr_conf_lpr_beta_beta/ + | /3 width=14 by cpr_conf_lpr_flat_theta/ + | /3 width=17 by cpr_conf_lpr_theta_theta/ + ] +] +qed-. + +(* Basic_1: includes: pr0_confluence pr2_confluence *) +theorem cpr_conf: ∀L. confluent … (cpr L). +/2 width=6 by cpr_conf_lpr/ qed-. + +(* Properties on context-sensitive parallel reduction for terms *************) + +lemma lpr_cpr_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 (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ +qed-. + +lemma lpr_cpr_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 (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ +qed-. (* Main properties **********************************************************)