X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flexs_lexs.ma;h=753b43b2efdc671f3325d2d7228734e7e685d0c0;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=7e7c3debca3fefda7eb09d60ef299918e5fa23fa;hpb=075441b55fa8a6fa693a1c96ed60ab4d87c42a2d;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma index 7e7c3debc..753b43b2e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma @@ -13,56 +13,71 @@ (**************************************************************************) include "ground_2/relocation/rtmap_sand.ma". -include "ground_2/relocation/rtmap_sor.ma". -include "basic_2/relocation/lexs.ma". +include "basic_2/relocation/drops.ma". (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****) (* Main properties **********************************************************) -theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f): - lexs_transitive RN1 RN2 RN RN1 RP1 → - lexs_transitive RP1 RP2 RP RN1 RP1 → - ∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 → - ∀L2. L0 ⦻*[RN2, RP2, f] L2 → - L1 ⦻*[RN, RP, f] L2. -#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0 -[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 // -| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H - #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/ -| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H - #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/ +theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP): + ∀L1,f. + (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → lexs_transitive RN1 RN2 RN RN1 RP1 g K I) → + (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → lexs_transitive RP1 RP2 RP RN1 RP1 g K I) → + ∀L0. L1 ⪤*[RN1, RP1, f] L0 → + ∀L2. L0 ⪤*[RN2, RP2, f] L2 → + L1 ⪤*[RN, RP, f] L2. +#RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1 +[ #f #_ #_ #L0 #H1 #L2 #H2 + lapply (lexs_inv_atom1 … H1) -H1 #H destruct + lapply (lexs_inv_atom1 … H2) -H2 #H destruct + /2 width=1 by lexs_atom/ +| #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct + #HN #HP #L0 #H1 #L2 #H2 + [ elim (lexs_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct + elim (lexs_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct + lapply (HP … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12 + lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_push, drops_drop/ + | elim (lexs_inv_next1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct + elim (lexs_inv_next1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct + lapply (HN … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12 + lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_next, drops_drop/ + ] ] qed-. -(* Basic_2A1: includes: lpx_sn_trans *) -theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RN RP → - lexs_transitive RP RP RP RN RP → +theorem lexs_trans (RN) (RP) (f): (∀g,I,K. lexs_transitive RN RN RN RN RP g K I) → + (∀g,I,K. lexs_transitive RP RP RP RN RP g K I) → Transitive … (lexs RN RP f). /2 width=9 by lexs_trans_gen/ qed-. -(* Basic_2A1: includes: lpx_sn_conf *) -theorem lexs_conf (RN1) (RP1) (RN2) (RP2): lexs_confluent RN1 RN2 RN1 RP1 RN2 RP2 → - lexs_confluent RP1 RP2 RN1 RP1 RN2 RP2 → - ∀f. confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f). -#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L0 -generalize in match f; -f elim L0 -L0 -[ #f #L1 #HL01 #L2 #HL02 -HRN -HRP - lapply (lexs_inv_atom1 … HL01) -HL01 #H destruct - lapply (lexs_inv_atom1 … HL02) -HL02 #H destruct - /2 width=3 by ex2_intro/ -| #K0 #I #V0 #IH #f #L1 #HL01 #L2 #HL02 - elim (pn_split f) * #g #H destruct - [ elim (lexs_inv_push1 … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct - elim (lexs_inv_push1 … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct - elim (IH … HK01 … HK02) -IH #K #HK1 #HK2 - elim (HRP … HV01 … HV02 … HK01 … HK02) -HRP -HRN -K0 -V0 - /3 width=5 by lexs_push, ex2_intro/ - | elim (lexs_inv_next1 … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct - elim (lexs_inv_next1 … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct - elim (IH … HK01 … HK02) -IH #K #HK1 #HK2 - elim (HRN … HV01 … HV02 … HK01 … HK02) -HRN -HRP -K0 -V0 - /3 width=5 by lexs_next, ex2_intro/ +theorem lexs_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤*[R1, cfull, f] L → 𝐈⦃f⦄ → + ∀L2. L ⪤*[R2, cfull, f] L2 → L1 ⪤*[R3, cfull, f] L2. +#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f +[ #f #Hf #L2 #H >(lexs_inv_atom1 … H) -L2 // ] +#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H +[ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf +elim (lexs_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct +/3 width=1 by lexs_push/ +qed-. + +theorem lexs_conf (RN1) (RP1) (RN2) (RP2): + ∀L,f. + (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) → + (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) → + pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L. +#RN1 #RP1 #RN2 #RP2 #L elim L -L +[ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1 + /2 width=3 by lexs_atom, ex2_intro/ +| #L #I0 #IH #f elim (pn_split f) * #g #H destruct + #HN #HP #Y1 #H1 #Y2 #H2 + [ elim (lexs_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct + elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct + elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2 + elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/ + | elim (lexs_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct + elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct + elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2 + elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/ ] ] qed-. @@ -78,11 +93,11 @@ theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) → /3 width=3 by/ qed-. lemma lexs_meet: ∀RN,RP,L1,L2. - ∀f1. L1 ⦻*[RN, RP, f1] L2 → - ∀f2. L1 ⦻*[RN, RP, f2] L2 → - ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2. + ∀f1. L1 ⪤*[RN, RP, f1] L2 → + ∀f2. L1 ⪤*[RN, RP, f2] L2 → + ∀f. f1 ⋒ f2 ≘ f → L1 ⪤*[RN, RP, f] L2. #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // -#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf +#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H [ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf) @@ -91,11 +106,11 @@ try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H qed-. lemma lexs_join: ∀RN,RP,L1,L2. - ∀f1. L1 ⦻*[RN, RP, f1] L2 → - ∀f2. L1 ⦻*[RN, RP, f2] L2 → - ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2. + ∀f1. L1 ⪤*[RN, RP, f1] L2 → + ∀f2. L1 ⪤*[RN, RP, f2] L2 → + ∀f. f1 ⋓ f2 ≘ f → L1 ⪤*[RN, RP, f] L2. #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // -#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf +#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H [ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)