X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fi_static%2Frexs.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fi_static%2Frexs.ma;h=e8bd0a2bd23b6e327afce277765345e929255db4;hb=222044da28742b24584549ba86b1805a87def070;hp=0000000000000000000000000000000000000000;hpb=5c186c72f508da0849058afeecc6877cd9ed6303;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/rexs.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/rexs.ma new file mode 100644 index 000000000..e8bd0a2bd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/i_static/rexs.ma @@ -0,0 +1,195 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground_2/lib/star.ma". +include "basic_2/notation/relations/relationstar_4.ma". +include "basic_2/static/rex.ma". + +(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***) + +definition rexs (R): term → relation lenv ≝ CTC … (rex R). + +interpretation "iterated extension on referred entries (local environment)" + 'RelationStar R T L1 L2 = (rexs R T L1 L2). + +(* Basic properties *********************************************************) + +lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L → + ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2. +#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *) +qed-. + +lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L → + ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2. +#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *) +qed-. + +lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆. +/2 width=1 by inj/ qed. + +lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s. + L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2 +/3 width=4 by rex_sort, rexs_step_dx, inj/ +qed. + +lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) → + ∀I,L1,L2,V. L1 ⪤*[R, V] L2 → + L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V. +#R #HR #I #L1 #L2 #V #H elim H -L2 +/3 width=5 by rex_pair, rexs_step_dx, inj/ +qed. + +lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 → + L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}. +/3 width=3 by rex_unit, inj/ qed. + +lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i. + L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2 +/3 width=4 by rex_lref, rexs_step_dx, inj/ +qed. + +lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l. + L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2 +/3 width=4 by rex_gref, rexs_step_dx, inj/ +qed. + +lemma rexs_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 #H elim H -L2 +/4 width=5 by rex_co, rexs_step_dx, inj/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *) +lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆. +#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/ +qed-. + +(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *) +lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆. +#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1 +/3 width=3 by inj, rex_inv_atom_dx/ +qed-. + +lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 → + ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ + | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 & + Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. +#R #Y1 #Y2 #s #H elim H -Y2 +[ #Y2 #H elim (rex_inv_sort … H) -H * + /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/ +| #Y #Y2 #_ #H elim (rex_inv_sort … H) -H * + [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/ + | #I #I2 #L #L2 #HL2 #H #H2 * * + [ #H1 #H0 destruct + | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct + /4 width=7 by ex3_4_intro, rexs_step_dx, or_intror/ + ] + ] +] +qed-. + +lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 → + ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ + | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 & + Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. +#R #Y1 #Y2 #l #H elim H -Y2 +[ #Y2 #H elim (rex_inv_gref … H) -H * + /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/ +| #Y #Y2 #_ #H elim (rex_inv_gref … H) -H * + [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/ + | #I #I2 #L #L2 #HL2 #H #H2 * * + [ #H1 #H0 destruct + | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct + /4 width=7 by ex3_4_intro, rexs_step_dx, or_intror/ + ] + ] +] +qed-. + +lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) → + ∀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 H -L2 +[ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/ +| #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/ +] +qed-. + +lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → + ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2. +#R #I #L1 #L2 #V #T #H elim H -L2 +[ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/ +| #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 → + ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}. +#R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H * +[ #H destruct +| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} → + ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}. +#R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H * +[ #_ #H destruct +| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 → + ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}. +#R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H * +[ #H destruct +| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} → + ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}. +#R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H * +[ #_ #H destruct +| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2. +#R #I #L1 #L2 #V #T #H elim H -L2 +/3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/ +qed-. + +lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) → + ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 → + L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V. +#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H // +qed-. + +lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2. +#R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H // +qed-. + +(* Basic_2A1: removed theorems 2: + TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2 +*)