X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Flpr_lpr.ma;h=284fafe2035b697917901c6a2e24cf80c77aed09;hb=499eb9f9a3107c6ee4edd7b6830a1cc1b054bbe2;hp=addc2f948cac5d52295872e1508e9252c8350936;hpb=7ad8c044ab33ea0f2aebb1c40fa20340d7f2f3eb;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 addc2f948..284fafe20 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 +12,9 @@ (* *) (**************************************************************************) -include "basic_2/grammar/lpx_sn_lpx_sn.ma". -include "basic_2/substitution/fqup.ma". -include "basic_2/reduction/lpr_ldrop.ma". +include "basic_2/substitution/lpx_sn_lpx_sn.ma". +include "basic_2/multiple/fqup.ma". +include "basic_2/reduction/lpr_drop.ma". (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) @@ -22,64 +22,62 @@ include "basic_2/reduction/lpr_ldrop.ma". fact cpr_conf_lpr_atom_atom: ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T. -/2 width=3/ qed-. +/2 width=3 by cpr_atom, ex2_intro/ qed-. fact cpr_conf_lpr_atom_delta: ∀G,L0,i. ( - ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀K0,V0. ⬇[i] L0 ≡ K0.ⓓV0 → + ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⬆[O, i + 1] V2 ≡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_drop_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_drop_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 (drop_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)) #T #HVT -lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ +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_lpr_delta_delta: ∀G,L0,i. ( - ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → - ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → - ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀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⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +lapply (drop_mono … H … HLK0) -H #H destruct +elim (lpr_drop_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 +lapply (drop_fwd_drop2 … HLK1) -W1 #HLK1 +elim (lpr_drop_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 (drop_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)) #T #HVT -lapply (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 -lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ +elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/ qed-. fact cpr_conf_lpr_bind_bind: ∀a,I,G,L0,V0,T0. ( - ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 @@ -91,48 +89,49 @@ fact cpr_conf_lpr_bind_bind: #a #I #G #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/ +elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH +/3 width=5 by lpr_pair, cpr_bind, ex2_intro/ qed-. fact cpr_conf_lpr_bind_zeta: ∀G,L0,V0,T0. ( - ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ 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 → + ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⬆[O, 1] X2 ≡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (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/ +elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -V0 -T0 #T #HT1 #HT2 +elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, drop_drop, ex2_intro/ qed-. fact cpr_conf_lpr_zeta_zeta: ∀G,L0,V0,T0. ( - ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ 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 → + ∀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⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (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/ +elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -T0 #T #HT1 #HT2 +elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by drop_drop/ #T1 #HT1 #HXT1 +elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by drop_drop/ #T2 #HT2 #HXT2 +lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/ qed-. fact cpr_conf_lpr_flat_flat: ∀I,G,L0,V0,T0. ( - ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 @@ -144,12 +143,12 @@ fact cpr_conf_lpr_flat_flat: #I #G #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/ +elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/ qed-. -fact cpr_conf_lpr_flat_tau: +fact cpr_conf_lpr_flat_eps: ∀G,L0,V0,T0. ( - ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 @@ -159,12 +158,12 @@ fact cpr_conf_lpr_flat_tau: ∃∃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 (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_eps, ex2_intro/ qed-. -fact cpr_conf_lpr_tau_tau: +fact cpr_conf_lpr_eps_eps: ∀G,L0,V0,T0. ( - ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 @@ -174,12 +173,12 @@ fact cpr_conf_lpr_tau_tau: ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T. #G #L0 #V0 #T0 #IH #T1 #HT01 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/ +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3 by ex2_intro/ qed-. fact cpr_conf_lpr_flat_beta: ∀a,G,L0,V0,W0,T0. ( - ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 @@ -191,11 +190,11 @@ fact cpr_conf_lpr_flat_beta: #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 (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ #W #HW1 #HW2 -elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 -lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1/ -/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/ (**) (* auto too slow without trace *) +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2 +elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *) +/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/ qed-. (* Basic-1: includes: @@ -204,36 +203,36 @@ qed-. *) fact cpr_conf_lpr_flat_theta: ∀a,G,L0,V0,W0,T0. ( - ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ 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 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⬆[O, 1] V2 ≡ U2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HV01 … HV02 … HL01 … HL02) -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=1/ #HU2 +lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/ #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 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/ (**) (* timeout=35 *) + elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ + elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -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=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 *) + elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 + elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 + /4 width=9 by cpr_flat, cpr_zeta, drop_drop, lift_flat, ex2_intro/ ] qed-. fact cpr_conf_lpr_beta_beta: ∀a,G,L0,V0,W0,T0. ( - ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 @@ -244,41 +243,41 @@ fact cpr_conf_lpr_beta_beta: ∃∃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 (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ #W #HW1 #HW2 -elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 -lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1/ -lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1/ -/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ +elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/ +lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ +/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *) qed-. (* Basic_1: was: pr0_upsilon_upsilon *) fact cpr_conf_lpr_theta_theta: ∀a,G,L0,V0,W0,T0. ( - ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ → + ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 ) → - ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 → + ∀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 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⬆[O, 1] V2 ≡ U2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (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 (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ +elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -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 by cpr_bind, cpr_flat, ex2_intro/ (**) (* timeout 40 *) +lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by drop_drop/ +lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/ +/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *) qed-. theorem cpr_conf_lpr: ∀G. lpx_sn_confluent (cpr G) (cpr G). -#G #L0 #T0 @(fqup_wf_ind … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [|*] +#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 @@ -320,11 +319,11 @@ theorem cpr_conf_lpr: ∀G. lpx_sn_confluent (cpr G) (cpr G). |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=8 by ex2_commute, cpr_conf_lpr_flat_eps/ | /4 width=12 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=8 by cpr_conf_lpr_flat_eps/ + | /3 width=7 by cpr_conf_lpr_eps_eps/ | /3 width=12 by cpr_conf_lpr_flat_beta/ | /3 width=13 by cpr_conf_lpr_beta_beta/ | /3 width=14 by cpr_conf_lpr_flat_theta/ @@ -342,13 +341,13 @@ theorem cpr_conf: ∀G,L. confluent … (cpr G L). lemma lpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T. #G #L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ +elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/ qed-. lemma lpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T. #G #L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ +elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/ qed-. (* Main properties **********************************************************)