X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfxs_lfxs.ma;h=c1d81666830090ea87a3f86218fba2f28a23f708;hb=1a590671c8e8551b01a6831843a22c9485d90511;hp=3b94bdb61dcca96f708ce6c416d8b1d212d51818;hpb=632c54beaf67e68a1eeeec22274466157003b779;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 3b94bdb61..c1d816668 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -13,18 +13,31 @@ (**************************************************************************) 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/lfxs.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) (* Advanced properties ******************************************************) +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-. + +(* 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). +#R #HR #L1 #L2 #T +elim (frees_total L1 T) #f #Hf +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. + lexs_frees_confluent … (cext2 R1) cfull → + ∀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 @@ -34,48 +47,98 @@ 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 /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. + lexs_frees_confluent … (cext2 R1) cfull → + ∀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 /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)) → + lexs_frees_confluent … (cext2 R1) cfull → + ∀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 (HR … Hf … H0) -HR -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)) → + lexs_frees_confluent … (cext2 R1) cfull → + ∀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 (HR … Hf … H0) -HR -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 → + lexs_frees_confluent (cext2 R1) cfull → + lexs_frees_confluent (cext2 R2) cfull → 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 @@ -88,10 +151,41 @@ elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ] 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-. + +(* Negated inversion lemmas *************************************************) + +(* 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 → ⊥). +#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 → ⊥). +#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-.