X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfxs.ma;h=a1d50e69f243bd9c9ea175138630e9d3008e16df;hb=56dd0e9f60e0dabfb587b014755fd4dad27960bb;hp=88a8bd542182ff72e19c288ded5cbe5915d3a725;hpb=984856dbab870ddc3156040df69b1f1846cc3aaf;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma index 88a8bd542..a1d50e69f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma @@ -14,7 +14,6 @@ include "ground_2/relocation/rtmap_id.ma". include "basic_2/notation/relations/relationstar_4.ma". -include "basic_2/grammar/ceq.ma". include "basic_2/relocation/lexs.ma". include "basic_2/static/frees.ma". @@ -26,14 +25,25 @@ definition lfxs (R) (T): relation lenv ≝ interpretation "generic extension on referred entries (local environment)" 'RelationStar R T L1 L2 = (lfxs R T L1 L2). -definition lfxs_confluent: relation4 (relation3 lenv term term) - (relation3 lenv term term) … ≝ - λR1,R2,RP1,RP2. - ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → - ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → - ∃∃T. R2 L1 T1 T & R1 L2 T2 T. +definition R_frees_confluent: predicate (relation3 lenv term term) ≝ + λRN. + ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 → + ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1. -(* Basic properties ***********************************************************) +definition lexs_frees_confluent: relation (relation3 lenv term term) ≝ + λRN,RP. + ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 → + ∀L2. L1 ⦻*[RN, RP, f1] L2 → + ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1. + +definition R_confluent2_lfxs: relation4 (relation3 lenv term term) + (relation3 lenv term term) … ≝ + λR1,R2,RP1,RP2. + ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → + ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → + ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + +(* Basic properties *********************************************************) lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. /3 width=3 by lexs_atom, frees_atom, ex2_intro/ @@ -67,6 +77,13 @@ lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. /3 width=5 by lexs_pair_repl, ex2_intro/ qed-. +lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull → + (∀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 (H1R … Hf1 … HL12) -Hf1 +/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/ +qed-. + lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2. #R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ @@ -82,8 +99,21 @@ lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆. #R #I #Y1 * /2 width=4 by lexs_inv_atom2/ qed-. +lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_sort … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct + /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 * @@ -94,7 +124,7 @@ lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → qed-. lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 * @@ -104,6 +134,19 @@ lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → ] qed-. +lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_gref … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct + /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf @@ -118,6 +161,22 @@ qed-. (* Advanced inversion lemmas ************************************************) +lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 → + ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 → ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & Y2 = L2.ⓑ{I}V2. @@ -154,6 +213,22 @@ lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 ] qed-. +lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 → + ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + (* Basic forward lemmas *****************************************************) lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.