X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfxs_lfxs.ma;h=f827a1f20a9f566ab4e2630b0fb5508e11662c30;hb=9323611e3819c1382b872a7ada00264991f36217;hp=78b1c526b359932c807b480cd7f903a91e6396ec;hpb=58ea181757dce19b875b2f5a224fe193b2263004;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 78b1c526b..f827a1f20 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -12,33 +12,64 @@ (* *) (**************************************************************************) +include "basic_2/relocation/lexs_length.ma". include "basic_2/relocation/lexs_lexs.ma". -include "basic_2/static/frees_fqup.ma". -include "basic_2/static/frees_frees.ma". +include "basic_2/static/frees_drops.ma". +include "basic_2/static/fle_fle.ma". include "basic_2/static/lfxs.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) -(* Advanced properties ******************************************************) +(* Advanced inversion lemmas ************************************************) -lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → - ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⦻*[R, cfull, f] L2. +lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → + ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⪤*[cext2 R, cfull, f] 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). + ∀L1,L2,T. Decidable (L1 ⪤*[R, T] L2). #R #HR #L1 #L2 #T elim (frees_total L1 T) #f #Hf -elim (lexs_dec R cfull HR … L1 L2 f) -/4 width=3 by lfxs_inv_frees, cfull_dec, ex2_intro, or_intror, or_introl/ +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)) → - lexs_frees_confluent … R1 cfull → - ∀L1,L2,V. L1 ⦻*[R1, V] L2 → ∀I,T. - ∃∃L. L1 ⦻*[R1, ②{I}V.T] L & L ⦻*[R2, V] L2. + 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 @@ -48,50 +79,98 @@ lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. refl 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 … HR1 HR2 … HL12 … Hfg) -HL12 #L #HL1 #HL2 +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 (HR … Hf … H) -HR -Hf -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)) → - lexs_frees_confluent … R1 cfull → - ∀L1,L2,T. L1 ⦻*[R1, T] L2 → ∀I,V. - ∃∃L. L1 ⦻*[R1, ⓕ{I}V.T] L & L ⦻*[R2, T] L2. + 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 … HR1 HR2 … HL12 … Hfg) -HL12 #L #HL1 #HL2 +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 (HR … Hf … H) -HR -Hf -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 *) theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T. - L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 → - L1 ⦻*[R, ⓑ{p,I}V1.T] L2. + L1 ⪤*[R, V1] L2 → L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2 → + L1 ⪤*[R, ⓑ{p,I}V1.T] L2. #R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 -elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2)) +lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2)) /3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/ qed. (* Basic_2A1: llpx_sn_flat *) theorem lfxs_flat: ∀R,I,L1,L2,V,T. - L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 → - L1 ⦻*[R, ⓕ{I}V.T] L2. + L1 ⪤*[R, V] L2 → L1 ⪤*[R, T] L2 → + L1 ⪤*[R, ⓕ{I}V.T] L2. #R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2) /3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/ qed. +theorem lfxs_bind_void: ∀R,p,I,L1,L2,V,T. + L1 ⪤*[R, V] L2 → L1.ⓧ ⪤*[R, T] L2.ⓧ → + L1 ⪤*[R, ⓑ{p,I}V.T] L2. +#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 +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. - lexs_frees_confluent R1 cfull → - lexs_frees_confluent R2 cfull → + 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 @@ -99,16 +178,75 @@ 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 (HR1 … Hf … HL01) -HL01 #f1 #Hf1 #H1 - elim (HR2 … Hf … HL02) -HL02 #f2 #Hf2 #H2 + 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 #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02 - elim (frees_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) /2 width=3 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-. @@ -116,16 +254,23 @@ qed-. (* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *) lemma lfnxs_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀p,I,L1,L2,V,T. (L1 ⦻*[R, ⓑ{p,I}V.T] L2 → ⊥) → - (L1 ⦻*[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V → ⊥). + ∀p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) → + (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V → ⊥). #R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V) /4 width=2 by lfxs_bind, or_intror, or_introl/ qed-. (* Basic_2A1: uses: nllpx_sn_inv_flat *) lemma lfnxs_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀I,L1,L2,V,T. (L1 ⦻*[R, ⓕ{I}V.T] L2 → ⊥) → - (L1 ⦻*[R, V] L2 → ⊥) ∨ (L1 ⦻*[R, T] L2 → ⊥). + ∀I,L1,L2,V,T. (L1 ⪤*[R, ⓕ{I}V.T] L2 → ⊥) → + (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1 ⪤*[R, T] L2 → ⊥). #R #HR #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V) /4 width=1 by lfxs_flat, or_intror, or_introl/ qed-. + +lemma lfnxs_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → + ∀p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) → + (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤*[R, T] L2.ⓧ → ⊥). +#R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V) +/4 width=2 by lfxs_bind_void, or_intror, or_introl/ +qed-.