X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Frdeq.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Frdeq.ma;h=0000000000000000000000000000000000000000;hb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;hp=efbd69ae18b78391821c794bec658258bf19bdaa;hpb=222044da28742b24584549ba86b1805a87def070;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/rdeq.ma deleted file mode 100644 index efbd69ae1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/rdeq.ma +++ /dev/null @@ -1,194 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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 "basic_2/notation/relations/stareqsn_5.ma". -include "basic_2/syntax/tdeq_ext.ma". -include "basic_2/static/rex.ma". - -(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) - -definition rdeq (h) (o): relation3 term lenv lenv ≝ - rex (cdeq h o). - -interpretation - "degree-based equivalence on referred entries (local environment)" - 'StarEqSn h o T L1 L2 = (rdeq h o T L1 L2). - -interpretation - "degree-based ranged equivalence (local environment)" - 'StarEqSn h o f L1 L2 = (sex (cdeq_ext h o) cfull f L1 L2). - -(* Basic properties ***********************************************************) - -lemma frees_tdeq_conf_rdeq (h) (o): ∀f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → ∀T2. T1 ≛[h, o] T2 → - ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≘ f. -#h #o #f #L1 #T1 #H elim H -f -L1 -T1 -[ #f #L1 #s1 #Hf #X #H1 #L2 #_ - elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct - /2 width=3 by frees_sort/ -| #f #i #Hf #X #H1 - >(tdeq_inv_lref1 … H1) -X #Y #H2 - >(sex_inv_atom1 … H2) -Y - /2 width=1 by frees_atom/ -| #f #I #L1 #V1 #_ #IH #X #H1 - >(tdeq_inv_lref1 … H1) -X #Y #H2 - elim (sex_inv_next1 … H2) -H2 #Z #L2 #HL12 #HZ #H destruct - elim (ext2_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct - /3 width=1 by frees_pair/ -| #f #I #L1 #Hf #X #H1 - >(tdeq_inv_lref1 … H1) -X #Y #H2 - elim (sex_inv_next1 … H2) -H2 #Z #L2 #_ #HZ #H destruct - >(ext2_inv_unit_sn … HZ) -Z /2 width=1 by frees_unit/ -| #f #I #L1 #i #_ #IH #X #H1 - >(tdeq_inv_lref1 … H1) -X #Y #H2 - elim (sex_inv_push1 … H2) -H2 #J #L2 #HL12 #_ #H destruct - /3 width=1 by frees_lref/ -| #f #L1 #l #Hf #X #H1 #L2 #_ - >(tdeq_inv_gref1 … H1) -X /2 width=1 by frees_gref/ -| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1 - elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct - /6 width=5 by frees_bind, sex_inv_tl, ext2_pair, sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn/ -| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1 - elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct - /5 width=5 by frees_flat, sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn/ -] -qed-. - -lemma frees_tdeq_conf (h) (o): ∀f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≘ f → - ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≘ f. -/4 width=7 by frees_tdeq_conf_rdeq, sex_refl, ext2_refl/ qed-. - -lemma frees_rdeq_conf (h) (o): ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f → - ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f. -/2 width=7 by frees_tdeq_conf_rdeq, tdeq_refl/ qed-. - -lemma tdeq_rex_conf (R) (h) (o): s_r_confluent1 … (cdeq h o) (rex R). -#R #h #o #L1 #T1 #T2 #HT12 #L2 * -/3 width=5 by frees_tdeq_conf, ex2_intro/ -qed-. - -lemma tdeq_rex_div (R) (h) (o): ∀T1,T2. T1 ≛[h, o] T2 → - ∀L1,L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2. -/3 width=5 by tdeq_rex_conf, tdeq_sym/ qed-. - -lemma tdeq_rdeq_conf (h) (o): s_r_confluent1 … (cdeq h o) (rdeq h o). -/2 width=5 by tdeq_rex_conf/ qed-. - -lemma tdeq_rdeq_div (h) (o): ∀T1,T2. T1 ≛[h, o] T2 → - ∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2. -/2 width=5 by tdeq_rex_div/ qed-. - -lemma rdeq_atom (h) (o): ∀I. ⋆ ≛[h, o, ⓪{I}] ⋆. -/2 width=1 by rex_atom/ qed. - -lemma rdeq_sort (h) (o): ∀I1,I2,L1,L2,s. - L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}. -/2 width=1 by rex_sort/ qed. - -lemma rdeq_pair (h) (o): ∀I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 → - L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2. -/2 width=1 by rex_pair/ qed. -(* -lemma rdeq_unit (h) (o): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext h o, cfull, f] L2 → - L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}. -/2 width=3 by rex_unit/ qed. -*) -lemma rdeq_lref (h) (o): ∀I1,I2,L1,L2,i. - L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #↑i] L2.ⓘ{I2}. -/2 width=1 by rex_lref/ qed. - -lemma rdeq_gref (h) (o): ∀I1,I2,L1,L2,l. - L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}. -/2 width=1 by rex_gref/ qed. - -lemma rdeq_bind_repl_dx (h) (o): ∀I,I1,L1,L2.∀T:term. - L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} → - ∀I2. I ≛[h, o] I2 → - L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}. -/2 width=2 by rex_bind_repl_dx/ qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma rdeq_inv_atom_sn (h) (o): ∀Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆. -/2 width=3 by rex_inv_atom_sn/ qed-. - -lemma rdeq_inv_atom_dx (h) (o): ∀Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆. -/2 width=3 by rex_inv_atom_dx/ qed-. -(* -lemma rdeq_inv_zero (h) (o): ∀Y1,Y2. Y1 ≛[h, o, #0] Y2 → - ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2 - | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o, cfull, f] L2 & - Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}. -#h #o #Y1 #Y2 #H elim (rex_inv_zero … H) -H * -/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/ -qed-. -*) -lemma rdeq_inv_lref (h) (o): ∀Y1,Y2,i. Y1 ≛[h, o, #↑i] Y2 → - ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 & - Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. -/2 width=1 by rex_inv_lref/ qed-. - -(* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *) -lemma rdeq_inv_bind (h) (o): ∀p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 → - ∧∧ L1 ≛[h, o, V] L2 & L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V. -/2 width=2 by rex_inv_bind/ qed-. - -(* Basic_2A1: uses: lleq_inv_flat *) -lemma rdeq_inv_flat (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → - ∧∧ L1 ≛[h, o, V] L2 & L1 ≛[h, o, T] L2. -/2 width=2 by rex_inv_flat/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma rdeq_inv_zero_pair_sn (h) (o): ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 → - ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2. -/2 width=1 by rex_inv_zero_pair_sn/ qed-. - -lemma rdeq_inv_zero_pair_dx (h) (o): ∀I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1. -/2 width=1 by rex_inv_zero_pair_dx/ qed-. - -lemma rdeq_inv_lref_bind_sn (h) (o): ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #↑i] Y2 → - ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}. -/2 width=2 by rex_inv_lref_bind_sn/ qed-. - -lemma rdeq_inv_lref_bind_dx (h) (o): ∀I2,Y1,L2,i. Y1 ≛[h, o, #↑i] L2.ⓘ{I2} → - ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}. -/2 width=2 by rex_inv_lref_bind_dx/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma rdeq_fwd_zero_pair (h) (o): ∀I,K1,K2,V1,V2. - K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2. -/2 width=3 by rex_fwd_zero_pair/ qed-. - -(* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *) -lemma rdeq_fwd_pair_sn (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[h, o, V] L2. -/2 width=3 by rex_fwd_pair_sn/ qed-. - -(* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *) -lemma rdeq_fwd_bind_dx (h) (o): ∀p,I,L1,L2,V,T. - L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V. -/2 width=2 by rex_fwd_bind_dx/ qed-. - -(* Basic_2A1: uses: lleq_fwd_flat_dx *) -lemma rdeq_fwd_flat_dx (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2. -/2 width=3 by rex_fwd_flat_dx/ qed-. - -lemma rdeq_fwd_dx (h) (o): ∀I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} → - ∃∃I1,K1. L1 = K1.ⓘ{I1}. -/2 width=5 by rex_fwd_dx/ qed-.