X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfxs_lfxs.ma;h=3b5982899d7ab94bfa1d4e106848f0702d0bccbf;hb=f7296f9cf2ee73465a374942c46b138f35c42ccb;hp=f827a1f20a9f566ab4e2630b0fb5508e11662c30;hpb=9323611e3819c1382b872a7ada00264991f36217;p=helm.git 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 f827a1f20..3b5982899 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -12,10 +12,8 @@ (* *) (**************************************************************************) -include "basic_2/relocation/lexs_length.ma". include "basic_2/relocation/lexs_lexs.ma". -include "basic_2/static/frees_drops.ma". -include "basic_2/static/fle_fle.ma". +include "basic_2/static/frees_fqup.ma". include "basic_2/static/lfxs.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) @@ -27,36 +25,8 @@ lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → #R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/ qed-. -lemma frees_lexs_conf: ∀R. lfxs_fle_compatible R → - ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 → - ∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 → - ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1. -#R #HR #L1 #T #f1 #Hf1 #L2 #H1L -lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L -@(fle_frees_trans_eq … H2L … Hf1) /3 width=4 by lexs_fwd_length, sym_eq/ -qed-. - -(* Properties with free variables inclusion for restricted closures *********) - -(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *) -lemma fle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ → - ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2. -#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12 -elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct -/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/ -qed-. - (* Advanced properties ******************************************************) -lemma lfxs_sym: ∀R. lfxs_fle_compatible R → - (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) → - ∀T. symmetric … (lfxs R T). -#R #H1R #H2R #T #L1 #L2 -* #f1 #Hf1 #HL12 -elim (frees_lexs_conf … Hf1 … HL12) -Hf1 // -/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/ -qed-. - (* Basic_2A1: uses: llpx_sn_dec *) lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → ∀L1,L2,T. Decidable (L1 ⪤*[R, T] L2). @@ -66,81 +36,6 @@ elim (lexs_dec (cext2 R) cfull … L1 L2 f) /4 width=3 by lfxs_inv_frees, cfull_dec, ext2_dec, ex2_intro, or_intror, or_introl/ qed-. -lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → - lfxs_fle_compatible R1 → - ∀L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T. - ∃∃L. L1 ⪤*[R1, ②{I}V.T] L & L ⪤*[R2, V] L2. -#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T -[ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg - elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy -| elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg - elim (frees_inv_flat … Hg) #y1 #y2 #H #_ #Hy -] -lapply(frees_mono … H … Hf) -H #H1 -lapply (sor_eq_repl_back1 … Hy … H1) -y1 #Hy -lapply (sor_inv_sle_sn … Hy) -y2 #Hfg -elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2 -lapply (sle_lexs_trans … HL1 … Hfg) // #H -elim (frees_lexs_conf … Hf … H) -Hf -H -/4 width=7 by sle_lexs_trans, ex2_intro/ -qed-. - -lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → - lfxs_fle_compatible R1 → - ∀L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V. - ∃∃L. L1 ⪤*[R1, ⓕ{I}V.T] L & L ⪤*[R2, T] L2. -#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V -elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg -elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy -lapply(frees_mono … H … Hf) -H #H2 -lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy -lapply (sor_inv_sle_dx … Hy) -y1 #Hfg -elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2 -lapply (sle_lexs_trans … HL1 … Hfg) // #H -elim (frees_lexs_conf … Hf … H) -Hf -H -/4 width=7 by sle_lexs_trans, ex2_intro/ -qed-. - -lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → - lfxs_fle_compatible R1 → - ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤*[R1, T] L2 → ∀p. - ∃∃L,V. L1 ⪤*[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤*[R2, T] L2 & R1 L1 V1 V. -#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p -elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg -elim (frees_inv_bind … Hg) #y1 #y2 #_ #H #Hy -lapply(frees_mono … H … Hf) -H #H2 -lapply (tl_eq_repl … H2) -H2 #H2 -lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy -lapply (sor_inv_sle_dx … Hy) -y1 #Hfg -lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg -elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2 -lapply (sle_lexs_trans … H … Hfg) // #H0 -elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H -elim (ext2_inv_pair_sn … H) -H #V #HV #H1 #H2 destruct -elim (frees_lexs_conf … Hf … H0) -Hf -H0 -/4 width=7 by sle_lexs_trans, ex3_2_intro, ex2_intro/ -qed-. - -lemma lfxs_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → - lfxs_fle_compatible R1 → - ∀L1,L2,T. L1.ⓧ ⪤*[R1, T] L2 → ∀p,I,V. - ∃∃L. L1 ⪤*[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤*[R2, T] L2. -#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V -elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg -elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy -lapply(frees_mono … H … Hf) -H #H2 -lapply (tl_eq_repl … H2) -H2 #H2 -lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy -lapply (sor_inv_sle_dx … Hy) -y1 #Hfg -lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg -elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2 -lapply (sle_lexs_trans … H … Hfg) // #H0 -elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H -elim (ext2_inv_unit_sn … H) -H #H destruct -elim (frees_lexs_conf … Hf … H0) -Hf -H0 -/4 width=7 by sle_lexs_trans, ex2_intro/ (* note: 2 ex2_intro *) -qed-. - (* Main properties **********************************************************) (* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *) @@ -168,88 +63,6 @@ lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2)) /3 width=7 by frees_fwd_isfin, frees_bind_void, lexs_join, isfin_tl, ex2_intro/ qed. -theorem lfxs_conf: ∀R1,R2. - lfxs_fle_compatible R1 → - lfxs_fle_compatible R2 → - R_confluent2_lfxs R1 R2 R1 R2 → - ∀T. confluent2 … (lfxs R1 T) (lfxs R2 T). -#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02 -lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12 -lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01 -elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ] -[ #L #HL1 #HL2 - elim (frees_lexs_conf … Hf … HL01) // -HR1 -HL01 #f1 #Hf1 #H1 - elim (frees_lexs_conf … Hf … HL02) // -HR2 -HL02 #f2 #Hf2 #H2 - lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1 - lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2 - /3 width=5 by ex2_intro/ -| #g * #I0 [2: #V0 ] #K0 #n #HLK0 #Hgf #Z1 #H1 #Z2 #H2 #K1 #HK01 #K2 #HK02 - [ elim (ext2_inv_pair_sn … H1) -H1 #V1 #HV01 #H destruct - elim (ext2_inv_pair_sn … H2) -H2 #V2 #HV02 #H destruct - elim (frees_inv_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0 - lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01 - lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02 - elim (HR12 … HV01 … HV02 K1 … K2) /3 width=3 by ext2_pair, ex2_intro/ - | lapply (ext2_inv_unit_sn … H1) -H1 #H destruct - lapply (ext2_inv_unit_sn … H2) -H2 #H destruct - /3 width=3 by ext2_unit, ex2_intro/ - ] -] -qed-. - -theorem lfxs_trans_gen: ∀R1,R2,R3. - c_reflexive … R1 → c_reflexive … R2 → - lfxs_confluent R1 R2 → lfxs_transitive R1 R2 R3 → - ∀L1,T,L. L1 ⪤*[R1, T] L → - ∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2. -#R1 #R2 #R3 #H1R #H2R #H3R #H4R #L1 #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L1 T) -L1 -T -#G0 #L0 #T0 #IH #G #L1 * * -[ #s #HG #HL #HT #L #H1 #L2 #H2 destruct - elim (lfxs_inv_sort … H1) -H1 * - [ #H1 #H0 destruct - >(lfxs_inv_atom_sn … H2) -L2 // - | #I1 #I #K1 #K #HK1 #H1 #H0 destruct - elim (lfxs_inv_sort_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct - /4 width=3 by lfxs_sort, fqu_fqup/ - ] -| * [ | #i ] #HG #HL #HT #L #H1 #L2 #H2 destruct - [ elim (lfxs_inv_zero … H1) -H1 * - [ #H1 #H0 destruct - >(lfxs_inv_atom_sn … H2) -L2 // - | #I #K1 #K #V1 #V #HK1 #H1 #H0 #H destruct - elim (lfxs_inv_zero_pair_sn … H2) -H2 #K2 #V2 #HK2 #HV2 #H destruct - /4 width=7 by lfxs_pair, fqu_fqup, fqu_lref_O/ - | #f1 #I #K1 #K #Hf1 #HK1 #H1 #H0 destruct - elim (lfxs_inv_zero_unit_sn … H2) -H2 #f2 #K2 #Hf2 #HK2 #H destruct - /5 width=8 by lfxs_unit, lexs_trans_id_cfull, lexs_eq_repl_back, isid_inv_eq_repl/ - ] - | elim (lfxs_inv_lref … H1) -H1 * - [ #H1 #H0 destruct - >(lfxs_inv_atom_sn … H2) -L2 // - | #I1 #I #K1 #K #HK1 #H1 #H0 destruct - elim (lfxs_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct - /4 width=3 by lfxs_lref, fqu_fqup/ - ] - ] -| #l #HG #HL #HT #L #H1 #L2 #H2 destruct - elim (lfxs_inv_gref … H1) -H1 * - [ #H1 #H0 destruct - >(lfxs_inv_atom_sn … H2) -L2 // - | #I1 #I #K1 #K #HK1 #H1 #H0 destruct - elim (lfxs_inv_gref_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct - /4 width=3 by lfxs_gref, fqu_fqup/ - ] -| #p #I #V1 #T1 #HG #HL #HT #L #H1 #L2 #H2 destruct - elim (lfxs_inv_bind … V1 V1 … H1) -H1 // #H1V #H1T - elim (lfxs_inv_bind … V1 V1 … H2) -H2 // #H2V #H2T - /3 width=4 by lfxs_bind/ -| #I #V1 #T1 #HG #HL #HT #L #H1 #L2 #H2 destruct - elim (lfxs_inv_flat … H1) -H1 #H1V #H1T - elim (lfxs_inv_flat … H2) -H2 #H2V #H2T - /3 width=3 by lfxs_flat/ -] -qed-. - (* Negated inversion lemmas *************************************************) (* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)