From: Ferruccio Guidi Date: Thu, 15 Sep 2016 15:28:39 +0000 (+0000) Subject: diamond property of reduction! X-Git-Tag: make_still_working~539 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=984856dbab870ddc3156040df69b1f1846cc3aaf;p=helm.git diamond property of reduction! --- diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma index 7fd35071a..3a779a649 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -181,6 +181,22 @@ lemma lexs_refl: ∀RN,RP,f. #L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/ qed. +lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2. + L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 → + ∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 → + L1.ⓑ{I}W1 ⦻*[RN, RP, f] L2.ⓑ{I}W2. +#RN #RP #f #I #L1 #L2 #V1 #V2 #HL12 #W1 #W2 #HN #HP +elim (lexs_fwd_pair … HL12) -HL12 /2 width=1 by lexs_inv_tl/ +qed-. + +lemma lexs_co: ∀RN1,RP1,RN2,RP2. + (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) → + (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) → + ∀f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2. +#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2 +/3 width=1 by lexs_atom, lexs_next, lexs_push/ +qed-. + lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) → ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 → ∀f1. f1 ⊆ f2 → L1 ⦻*[RN, RP, f1] L2. @@ -206,11 +222,3 @@ lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) → #g2 #H #H2 destruct /3 width=5 by lexs_next/ ] qed-. - -lemma lexs_co: ∀RN1,RP1,RN2,RP2. - (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) → - (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) → - ∀f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2. -#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2 -/3 width=1 by lexs_atom, lexs_next, lexs_push/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma index a5d3ca4d3..6bcfa173e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma @@ -107,7 +107,3 @@ qed-. pr2_gen_csort pr2_gen_cflat pr2_gen_cbind pr2_gen_ctail pr2_ctail *) -(* Basic_1: removed local theorems 4: - pr0_delta_eps pr0_cong_delta - pr2_free_free pr2_free_delta -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma index a38aff49d..299b0f8c8 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma @@ -46,6 +46,12 @@ lemma lfpr_gref: ∀h,I,G,L1,L2,V1,V2,l. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, §l] L2.ⓑ{I}V2. /2 width=1 by lfxs_gref/ qed. +lemma lfpr_pair_repl_dx: ∀h,I,G,L1,L2,T,V,V1. + ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V1 → + ∀V2. ⦃G, L1⦄ ⊢ V ➡[h] V2 → + ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V2. +/2 width=2 by lfxs_pair_repl_dx/ qed-. + (* Basic inversion lemmas ***************************************************) lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 = ⋆. @@ -117,14 +123,16 @@ lemma lfpr_fwd_pair_sn: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2. /2 width=3 by lfxs_fwd_pair_sn/ qed-. -(* Basic_2A1: removed theorems 11: +(* Basic_2A1: removed theorems 14: lpr_inv_atom1 lpr_inv_pair1 lpr_inv_atom2 lpr_inv_pair2 lpr_refl lpr_pair lpr_fwd_length lpr_lpx lpr_drop_conf drop_lpr_trans lpr_drop_trans_O1 + cpr_conf_lpr lpr_cpr_conf_dx lpr_cpr_conf_sn *) -(* Basic_1: removed theorems 7: wcpr0_gen_sort wcpr0_gen_head - wcpr0_getl wcpr0_getl_back - pr0_subst1_back - wcpr0_drop wcpr0_drop_back +(* Basic_1: removed theorems 7: + wcpr0_gen_sort wcpr0_gen_head + wcpr0_getl wcpr0_getl_back + pr0_subst1_back + wcpr0_drop wcpr0_drop_back *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_drops.ma index b3deebd85..3156e7252 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_drops.ma @@ -30,3 +30,11 @@ lemma lfpr_drops_conf: ∀h,G. dropable_sn (cpm 0 h G). lemma lfpr_drops_trans: ∀h,G. dropable_dx (cpm 0 h G). /2 width=5 by lfxs_dropable_dx/ qed-. + +lemma lfpr_inv_lref_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 → + ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2. +/2 width=3 by lfxs_inv_lref_sn/ qed-. + +lemma lfpr_inv_lref_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2. +/2 width=3 by lfxs_inv_lref_dx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_lfpr.ma new file mode 100644 index 000000000..e8955fd1e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr_lfpr.ma @@ -0,0 +1,383 @@ +(**************************************************************************) +(* ___ *) +(* ||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/rt_transition/cpm_lsubr.ma". +include "basic_2/rt_transition/cpr.ma". +include "basic_2/rt_transition/cpr_drops.ma". +include "basic_2/rt_transition/lfpr_drops.ma". +include "basic_2/rt_transition/lfpr_fqup.ma". + +(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************) + +(* Main properties with context-sensitive parallel r-transition for terms ***) + +fact cpr_conf_lfpr_atom_atom: + ∀h,I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T. +/2 width=3 by ex2_intro/ qed-. + +fact cpr_conf_lfpr_atom_delta: + ∀h,G,L0,i. ( + ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀K0,V0. ⬇*[i] L0 ≡ K0.ⓓV0 → + ∀V2. ⦃G, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[⫯i] V2 ≡ T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, #i] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, #i] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ #i ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T. +#h #G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_lref_sn … HL01 … HLK0) -HL01 #K1 #V1 #HLK1 #HK01 #HV01 +elim (lfpr_inv_lref_sn … HL02 … HLK0) -HL02 #K2 #W2 #HLK2 #HK02 #_ +lapply (drops_isuni_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 (cpm_lifts … HV2 … HLK2 … HVT2) -K2 -V2 +/3 width=6 by cpm_delta_drops, ex2_intro/ +qed-. + +fact cpr_conf_lfpr_delta_delta: + ∀h,G,L0,i. ( + ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀K0,V0. ⬇*[i] L0 ≡ K0.ⓓV0 → + ∀V1. ⦃G, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[⫯i] V1 ≡ T1 → + ∀KX,VX. ⬇*[i] L0 ≡ KX.ⓓVX → + ∀V2. ⦃G, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[⫯i] V2 ≡ T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, #i] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, #i] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T. +#h #G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 +#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +lapply (drops_mono … H … HLK0) -H #H destruct +elim (lfpr_inv_lref_sn … HL01 … HLK0) -HL01 #K1 #W1 #HLK1 #HK01 #_ +lapply (drops_isuni_fwd_drop2 … HLK1) -W1 // #HLK1 +elim (lfpr_inv_lref_sn … HL02 … HLK0) -HL02 #K2 #W2 #HLK2 #HK02 #_ +lapply (drops_isuni_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 (cpm_lifts … HV1 … HLK1 … HVT1) -K1 -V1 #T #HVT #HT1 +elim (cpm_lifts … HV2 … HLK2 … HVT2) -K2 -V2 #X #HX #HT2 +lapply (lifts_mono … HX … HVT) #H destruct +/2 width=3 by ex2_intro/ +qed-. + +fact cpr_conf_lfpr_bind_bind: + ∀h,p,I,G,L0,V0,T0. ( + ∀L,T. ⦃G, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓑ{p,I}V0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓑ{p,I}V0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T. +#h #p #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0 +elim (lfpr_inv_bind … 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 lfpr_pair_repl_dx, cpm_bind, ex2_intro/ +qed-. + +fact cpr_conf_lfpr_bind_zeta: + ∀h,G,L0,V0,T0. ( + ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → + ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≡ T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T. +#h #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0 +elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0 +elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=4 by lfpr_pair_repl_dx/ -L0 -V0 -T0 #T #HT1 #HT2 +elim (cpm_inv_lifts1 … HT2 … L2 … HXT2) -T2 /3 width=3 by drops_refl, drops_drop, cpm_zeta, ex2_intro/ +qed-. + +fact cpr_conf_lfpr_zeta_zeta: + ∀h,G,L0,V0,T0. ( + ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → ∀X1. ⬆*[1] X1 ≡ T1 → + ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≡ T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, +ⓓV0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T. +#h #G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_bind … HL01) -HL01 #H1V0 #H1T0 +elim (lfpr_inv_bind … HL02) -HL02 #H2V0 #H2T0 +elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=4 by lfpr_pair_repl_dx/ -L0 -T0 #T #HT1 #HT2 +elim (cpm_inv_lifts1 … HT1 … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #T1 #HT1 #HXT1 +elim (cpm_inv_lifts1 … HT2 … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #T2 #HT2 #HXT2 +lapply (lifts_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/ +qed-. + +fact cpr_conf_lfpr_flat_flat: + ∀h,I,G,L0,V0,T0. ( + ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓕ{I}V0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓕ{I}V0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T. +#h #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #H1T0 +elim (lfpr_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_lfpr_flat_epsilon: + ∀h,G,L0,V0,T0. ( + ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T. +#h #G #L0 #V0 #T0 #IH #V1 #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_flat … HL01) -HL01 #_ #H1T0 +elim (lfpr_inv_flat … HL02) -HL02 #_ #H2T0 +elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpm_eps, ex2_intro/ +qed-. + +fact cpr_conf_lfpr_epsilon_epsilon: + ∀h,G,L0,V0,T0. ( + ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓝV0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, L2⦄ ⊢ T2 ➡[h] T. +#h #G #L0 #V0 #T0 #IH #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_flat … HL01) -HL01 #_ #H1T0 +elim (lfpr_inv_flat … HL02) -HL02 #_ #H2T0 +elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/ +qed-. + +fact cpr_conf_lfpr_flat_beta: + ∀h,p,G,L0,V0,W0,T0. ( + ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T. +#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H +#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct +elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01 +elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0 +elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02 +elim (lfpr_inv_bind … 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 lfpr_pair_repl_dx/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *) +/4 width=5 by cpm_bind, cpr_flat, cpm_beta, ex2_intro/ +qed-. + +fact cpr_conf_lfpr_flat_theta: + ∀h,p,G,L0,V0,W0,T0. ( + ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≡ U2 → + ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T. +#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H +#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01 +elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0 +elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02 +elim (lfpr_inv_bind … HL02) -HL02 #H2W0 #H2T0 +elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 +elim (cpm_lifts … HV2 … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU2 +elim (cpm_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 lfpr_pair_repl_dx/ -L0 -V0 -W0 -T0 + /4 width=7 by cpm_bind, cpr_flat, cpm_theta, ex2_intro/ +| #T1 #HT01 #HXT1 #H destruct + elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=4 by lfpr_pair_repl_dx/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 + elim (cpm_inv_lifts1 … HT1 … L1 … HXT1) -HXT1 + /4 width=9 by cpr_flat, cpm_zeta, drops_refl, drops_drop, lifts_flat, ex2_intro/ +] +qed-. + +fact cpr_conf_lfpr_beta_beta: + ∀h,p,G,L0,V0,W0,T0. ( + ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓛ{p}W0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T. +#h #p #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01 +#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01 +elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0 +elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02 +elim (lfpr_inv_bind … 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 lfpr_pair_repl_dx/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (lsubr_cpm_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/ +lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ +/4 width=5 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *) +qed-. + +fact cpr_conf_lfpr_theta_theta: + ∀h,p,G,L0,V0,W0,T0. ( + ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0 + ) → + ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≡ U1 → + ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≡ U2 → + ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓐV0.ⓓ{p}W0.T0] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T. +#h #p #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 (lfpr_inv_flat … HL01) -HL01 #H1V0 #HL01 +elim (lfpr_inv_bind … HL01) -HL01 #H1W0 #H1T0 +elim (lfpr_inv_flat … HL02) -HL02 #H2V0 #HL02 +elim (lfpr_inv_bind … 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 lfpr_pair_repl_dx/ -L0 -V0 -W0 -T0 +elim (cpm_lifts … HV1 … (L1.ⓓW1) … HVU1) -HVU1 /3 width=2 by drops_refl, drops_drop/ #U #HVU +elim (cpm_lifts … HV2 … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #X #HX +lapply (lifts_mono … HX … HVU) -HX #H destruct +/4 width=7 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *) +qed-. + +theorem cpr_conf_lfpr: ∀h,G. lfxs_confluent (cpm 0 h G) (cpm 0 h G) (cpm 0 h G) (cpm 0 h G). +#h #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_drops … H1) -H1 + elim (cpr_inv_atom1_drops … H2) -H2 + [ #H2 #H1 destruct + /2 width=1 by cpr_conf_lfpr_atom_atom/ + | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct + /3 width=10 by cpr_conf_lfpr_atom_delta/ + | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct + /4 width=10 by ex2_commute, cpr_conf_lfpr_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_lfpr_delta_delta/ + ] +| #p #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpm_inv_bind1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #T1 #HT01 #HXT1 #H11 #H12 + ] + elim (cpm_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_lfpr_bind_bind/ + | /4 width=11 by ex2_commute, cpr_conf_lfpr_bind_zeta/ + | /3 width=11 by cpr_conf_lfpr_bind_zeta/ + | /3 width=12 by cpr_conf_lfpr_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 + | #p1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13 + | #p1 #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: #p2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23 + |4,8,12,16: #p2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23 + ] destruct + [ /3 width=10 by cpr_conf_lfpr_flat_flat/ + | /4 width=8 by ex2_commute, cpr_conf_lfpr_flat_epsilon/ + | /4 width=12 by ex2_commute, cpr_conf_lfpr_flat_beta/ + | /4 width=14 by ex2_commute, cpr_conf_lfpr_flat_theta/ + | /3 width=8 by cpr_conf_lfpr_flat_epsilon/ + | /3 width=8 by cpr_conf_lfpr_epsilon_epsilon/ + | /3 width=12 by cpr_conf_lfpr_flat_beta/ + | /3 width=13 by cpr_conf_lfpr_beta_beta/ + | /3 width=14 by cpr_conf_lfpr_flat_theta/ + | /3 width=17 by cpr_conf_lfpr_theta_theta/ + ] +] +qed-. + +(* Basic_1: includes: pr0_confluence pr2_confluence *) +theorem cpr_conf: ∀h,G,L. confluent … (cpm 0 h G L). +/2 width=6 by cpr_conf_lfpr/ qed-. + +(* Properties with context-sensitive parallel r-transition for terms ********) + +lemma lfpr_cpr_conf_dx: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, T0] L1 → + ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T. +#h #G #L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_conf_lfpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/ +qed-. + +lemma lfpr_cpr_conf_sn: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, T0] L1 → + ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T. +#h #G #L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_conf_lfpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/ +qed-. + +(* Main properties **********************************************************) + +(* + +theorem lpr_conf: ∀G. confluent … (lpr G). +/3 width=6 by lpx_sn_conf, cpr_conf_lpr/ +qed-. + +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx.ma index 57c0184be..c0ca76840 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx.ma @@ -46,6 +46,12 @@ lemma lfpx_gref: ∀h,I,G,L1,L2,V1,V2,l. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, §l] L2.ⓑ{I}V2. /2 width=1 by lfxs_gref/ qed. +lemma lfpx_pair_repl_dx: ∀h,I,G,L1,L2,T,V,V1. + ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V1 → + ∀V2. ⦃G, L1⦄ ⊢ V ⬈[h] V2 → + ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2. +/2 width=2 by lfxs_pair_repl_dx/ qed-. + (* Basic inversion lemmas ***************************************************) lemma lfpx_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] Y2 → Y2 = ⋆. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_drops.ma index 55c405162..086e71e87 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_drops.ma @@ -30,3 +30,11 @@ lemma lfpx_drops_conf: ∀h,G. dropable_sn (cpx h G). lemma lfpx_drops_trans: ∀h,G. dropable_dx (cpx h G). /2 width=5 by lfxs_dropable_dx/ qed-. + +lemma lfpx_inv_lref_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 → + ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ⬈[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2. +/2 width=3 by lfxs_inv_lref_sn/ qed-. + +lemma lfpx_inv_lref_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ⬈[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2. +/2 width=3 by lfxs_inv_lref_dx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma deleted file mode 100644 index fad8b481a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma +++ /dev/null @@ -1,357 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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/lpx_sn_lpx_sn.ma". -include "basic_2/multiple/fqup.ma". -include "basic_2/reduction/lpr_drop.ma". - -(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) - -(* Main properties on context-sensitive parallel reduction for terms ********) - -fact cpr_conf_lpr_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_lpr_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⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ 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⦄ ⊢ ➡ 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_drop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct -elim (lpr_drop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -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)) -/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⦄ → - ∀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. ⬇[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 (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 (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 (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)) /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⦄ → - ∀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.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 → - ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (IH … HV01 … HV02 … HL01 … HL02) // -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⦄ → - ∀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 → - ∀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 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⦄ → - ∀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 → - ∀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 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⦄ → - ∀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⦄ ⊢ T0 ➡ T1 → - ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/ -qed-. - -fact cpr_conf_lpr_flat_eps: - ∀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⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → - ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 - ) → - ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_eps, ex2_intro/ -qed-. - -fact cpr_conf_lpr_eps_eps: - ∀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⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 → - ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0 - ) → - ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 → - ∃∃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 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⦄ → - ∀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 → ∀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.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 (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: - pr0_cong_upsilon_refl pr0_cong_upsilon_zeta - pr0_cong_upsilon_cong pr0_cong_upsilon_delta -*) -fact cpr_conf_lpr_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⦄ ⊢ ➡ 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 → - ∀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 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 drop_drop/ #HU2 -elim (cpr_inv_abbr1 … H) -H * -[ #W1 #T1 #HW01 #HT01 #H destruct - 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 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⦄ → - ∀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 → ∀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⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ 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 (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 … 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⦄ → - ∀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 → - ∀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⦄ ⊢ ➡ 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 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=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_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_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 #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_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 #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_lpr_flat_flat/ - | /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_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/ - | /3 width=17 by cpr_conf_lpr_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_lpr/ qed-. - -(* Properties on context-sensitive parallel reduction for terms *************) - -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) /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) /2 width=3 by ex2_intro/ -qed-. - -(* Main properties **********************************************************) - -theorem lpr_conf: ∀G. confluent … (lpr G). -/3 width=6 by lpx_sn_conf, cpr_conf_lpr/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt index 63773c485..d0eb3451c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt @@ -3,4 +3,4 @@ cpx.ma cpx_simple.ma cpx_drops.ma cpx_lsubr.ma lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma cpm.ma cpm_simple.ma cpm_drops.ma cpm_lsubr.ma cpm_cpx.ma cpr.ma cpr_drops.ma -lfpr.ma lfpr_length.ma lfpr_drops.ma lfpr_fqup.ma lfpr_lfpx.ma +lfpr.ma lfpr_length.ma lfpr_drops.ma lfpr_fqup.ma lfpr_lfpx.ma lfpr_lfpr.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma index b3c7a7fe0..88a8bd542 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma @@ -26,6 +26,13 @@ definition lfxs (R) (T): relation lenv ≝ interpretation "generic extension on referred entries (local environment)" 'RelationStar R T L1 L2 = (lfxs R T L1 L2). +definition lfxs_confluent: relation4 (relation3 lenv term term) + (relation3 lenv term term) … ≝ + λR1,R2,RP1,RP2. + ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → + ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → + ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + (* Basic properties ***********************************************************) lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. @@ -52,6 +59,14 @@ lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l. #R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/ qed. +lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. + L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 → + ∀V2. R L1 V V2 → + L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2. +#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR +/3 width=5 by lexs_pair_repl, ex2_intro/ +qed-. + lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2. #R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma index 0a64c396b..f43eb4e95 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma @@ -78,3 +78,17 @@ lemma lfxs_inv_lift_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 → elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY lapply (drops_mono … HY … HLK2) -L2 -i #H destruct // qed-. + +lemma lfxs_inv_lref_sn: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 → + ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. +#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // +#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY +#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. + +lemma lfxs_inv_lref_dx: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. +#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // +#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY +#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma index ece33413e..5360aeddc 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -17,7 +17,7 @@ include "basic_2/static/lfxs.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) -(* Main properties ******************************************************) +(* Main properties **********************************************************) theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 → diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl index 5b07ac493..397134959 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl +++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl @@ -144,7 +144,7 @@ table { ] *) [ { "t-bound context-sensitive rt-transition" * } { - [ "lfpr ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fqup" + "lfpr_lfpx" * ] + [ "lfpr ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fqup" + "lfpr_lfpx" + "lfpr_lfpr" * ] [ "cpr ( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" (* + "cpr_llpx_sn" + "cpr_cir" *) * ] [ "cpm ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_drops" + "cpm_lsubr" + "cpm_cpx" * ] }