X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Fsex_sex.ma;h=a92cf28c1ead5f1a28274416a987a6ad6917e5f6;hb=dc605ae41c39773f55381f241b1ed3db4acf5edd;hp=239966181fd18c1e0adf4cc34c0f5b9e2de3b741;hpb=0af3592e3a85a4bb82c5c6df259cf9ab117ba0b1;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma index 239966181..a92cf28c1 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -include "ground_2/relocation/rtmap_sand.ma". +include "ground/relocation/rtmap_sand.ma". include "static_2/relocation/drops.ma". (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****) @@ -20,12 +20,12 @@ include "static_2/relocation/drops.ma". (* Main properties **********************************************************) theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP): - ∀L1,f. - (∀g,I,K,n. ⇩*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) → - (∀g,I,K,n. ⇩*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → sex_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. + ∀L1,f. + (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_transitive_sex RN1 RN2 RN RN1 RP1 g K I) → + (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_transitive_sex 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 (sex_inv_atom1 … H1) -H1 #H destruct @@ -45,13 +45,15 @@ theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP): ] qed-. -theorem sex_trans (RN) (RP) (f): (∀g,I,K. sex_transitive RN RN RN RN RP g K I) → - (∀g,I,K. sex_transitive RP RP RP RN RP g K I) → - Transitive … (sex RN RP f). +theorem sex_trans (RN) (RP) (f): + (∀g,I,K. R_pw_transitive_sex RN RN RN RN RP g K I) → + (∀g,I,K. R_pw_transitive_sex RP RP RP RN RP g K I) → + Transitive … (sex RN RP f). /2 width=9 by sex_trans_gen/ qed-. -theorem sex_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. +theorem sex_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 >(sex_inv_atom1 … H) -L2 // ] #f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H @@ -61,10 +63,10 @@ elim (sex_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct qed-. theorem sex_conf (RN1) (RP1) (RN2) (RP2): - ∀L,f. - (∀g,I,K,n. ⇩*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_sex RN1 RN2 RN1 RP1 RN2 RP2 g K I) → - (∀g,I,K,n. ⇩*[n] L ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) → - pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L. + ∀L,f. + (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_confluent2_sex RN1 RN2 RN1 RP1 RN2 RP2 g K I) → + (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) → + pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L. #RN1 #RP1 #RN2 #RP2 #L elim L -L [ #f #_ #_ #L1 #H1 #L2 #H2 >(sex_inv_atom1 … H1) >(sex_inv_atom1 … H2) -H2 -H1 /2 width=3 by sex_atom, ex2_intro/ @@ -82,20 +84,44 @@ theorem sex_conf (RN1) (RP1) (RN2) (RP2): ] qed-. -theorem sex_canc_sn: ∀RN,RP,f. Transitive … (sex RN RP f) → - symmetric … (sex RN RP f) → - left_cancellable … (sex RN RP f). +lemma sex_repl (RN) (RP) (SN) (SP) (L1) (f): + (∀g,I,K1,n. ⇩[n] L1 ≘ K1.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_replace3_sex … RN SN RN RP SN SP g K1 I) → + (∀g,I,K1,n. ⇩[n] L1 ≘ K1.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_replace3_sex … RP SP RN RP SN SP g K1 I) → + ∀L2. L1 ⪤[RN,RP,f] L2 → ∀K1. L1 ⪤[SN,SP,f] K1 → + ∀K2. L2 ⪤[SN,SP,f] K2 → K1 ⪤[RN,RP,f] K2. +#RN #RP #SN #SP #L1 elim L1 -L1 +[ #f #_ #_ #Y #HY #Y1 #HY1 #Y2 #HY2 + lapply (sex_inv_atom1 … HY) -HY #H destruct + lapply (sex_inv_atom1 … HY1) -HY1 #H destruct + lapply (sex_inv_atom1 … HY2) -HY2 #H destruct // +| #L1 #I1 #IH #f elim (pn_split f) * #g #H destruct + #HN #HP #Y #HY #Y1 #HY1 #Y2 #HY2 + [ elim (sex_inv_push1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct + elim (sex_inv_push1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct + elim (sex_inv_push1 … HY2) -HY2 #J2 #K2 #HLK2 #HIJ2 #H destruct + /5 width=13 by sex_push, drops_refl, drops_drop/ + | elim (sex_inv_next1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct + elim (sex_inv_next1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct + elim (sex_inv_next1 … HY2) -HY2 #J2 #K2 #HLK2 #HIJ2 #H destruct + /5 width=13 by sex_next, drops_refl, drops_drop/ + ] +] +qed-. + +theorem sex_canc_sn (RN) (RP): + ∀f. Transitive … (sex RN RP f) → symmetric … (sex RN RP f) → + left_cancellable … (sex RN RP f). /3 width=3 by/ qed-. -theorem sex_canc_dx: ∀RN,RP,f. Transitive … (sex RN RP f) → - symmetric … (sex RN RP f) → - right_cancellable … (sex RN RP f). +theorem sex_canc_dx (RN) (RP): + ∀f. Transitive … (sex RN RP f) → symmetric … (sex RN RP f) → + right_cancellable … (sex RN RP f). /3 width=3 by/ qed-. -lemma sex_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. +lemma sex_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. #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct @@ -105,10 +131,10 @@ try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H ] -Hf /3 width=5 by sex_next, sex_push/ qed-. -lemma sex_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. +lemma sex_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. #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct