X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfdeq.ma;h=4103c54c50ce4e191e7a55ce9b876df1e8ee52c5;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=8a770328f0758d0c063c21648c4058cabbb20915;hpb=6167cca50de37eba76a062537b24f7caef5b34f2;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma index 8a770328f..4103c54c5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma @@ -18,8 +18,8 @@ include "basic_2/static/lfxs.ma". (* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) -definition lfdeq: ∀h. sd h → relation3 term lenv lenv ≝ - λh,o. lfxs (cdeq h o). +definition lfdeq (h) (o): relation3 term lenv lenv ≝ + lfxs (cdeq h o). interpretation "degree-based equivalence on referred entries (local environment)" @@ -28,14 +28,11 @@ interpretation interpretation "degree-based ranged equivalence (local environment)" 'StarEqSn h o f L1 L2 = (lexs (cdeq_ext h o) cfull f L1 L2). -(* -definition lfdeq_transitive: predicate (relation3 lenv term term) ≝ - λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≛[h, o, T1] L2 → R L1 T1 T2. -*) + (* Basic properties ***********************************************************) -lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 → - ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f. +lemma frees_tdeq_conf_lfdeq (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 @@ -68,138 +65,130 @@ lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2 ] 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_lexs, lexs_refl, ext2_refl/ 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_lfdeq, lexs_refl, ext2_refl/ qed-. -lemma frees_lexs_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_lexs, tdeq_refl/ qed-. +lemma frees_lfdeq_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_lfdeq, tdeq_refl/ qed-. -lemma frees_lfdeq_conf_lexs: ∀h,o. lexs_frees_confluent (cdeq_ext h o) cfull. -/3 width=7 by frees_tdeq_conf_lexs, ex2_intro/ qed-. - -lemma tdeq_lfdeq_conf_sn: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o). -#h #o #L1 #T1 #T2 #HT12 #L2 * +lemma tdeq_lfxs_conf (R) (h) (o): s_r_confluent1 … (cdeq h o) (lfxs R). +#R #h #o #L1 #T1 #T2 #HT12 #L2 * /3 width=5 by frees_tdeq_conf, ex2_intro/ qed-. -(* Basic_2A1: uses: lleq_sym *) -lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T). -#h #o #T #L1 #L2 * -/4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/ -qed-. +lemma tdeq_lfxs_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_lfxs_conf, tdeq_sym/ qed-. -lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆. +lemma tdeq_lfdeq_conf (h) (o): s_r_confluent1 … (cdeq h o) (lfdeq h o). +/2 width=5 by tdeq_lfxs_conf/ qed-. + +lemma tdeq_lfdeq_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_lfxs_div/ qed-. + +lemma lfdeq_atom (h) (o): ∀I. ⋆ ≛[h, o, ⓪{I}] ⋆. /2 width=1 by lfxs_atom/ qed. -(* Basic_2A1: uses: lleq_sort *) -lemma lfdeq_sort: ∀h,o,I1,I2,L1,L2,s. - L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}. +lemma lfdeq_sort (h) (o): ∀I1,I2,L1,L2,s. + L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}. /2 width=1 by lfxs_sort/ qed. -lemma lfdeq_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. +lemma lfdeq_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 lfxs_pair/ qed. (* -lemma lfdeq_unit: ∀h,o,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cdeq_ext h o, cfull, f] L2 → - L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}. +lemma lfdeq_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 lfxs_unit/ qed. *) -lemma lfdeq_lref: ∀h,o,I1,I2,L1,L2,i. - L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #⫯i] L2.ⓘ{I2}. +lemma lfdeq_lref (h) (o): ∀I1,I2,L1,L2,i. + L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #↑i] L2.ⓘ{I2}. /2 width=1 by lfxs_lref/ qed. -(* Basic_2A1: uses: lleq_gref *) -lemma lfdeq_gref: ∀h,o,I1,I2,L1,L2,l. - L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}. +lemma lfdeq_gref (h) (o): ∀I1,I2,L1,L2,l. + L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}. /2 width=1 by lfxs_gref/ qed. -lemma lfdeq_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}. +lemma lfdeq_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 lfxs_bind_repl_dx/ qed-. (* Basic inversion lemmas ***************************************************) -lemma lfdeq_inv_atom_sn: ∀h,o,Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆. +lemma lfdeq_inv_atom_sn (h) (o): ∀Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆. /2 width=3 by lfxs_inv_atom_sn/ qed-. -lemma lfdeq_inv_atom_dx: ∀h,o,Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆. +lemma lfdeq_inv_atom_dx (h) (o): ∀Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆. /2 width=3 by lfxs_inv_atom_dx/ qed-. (* -lemma lfdeq_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}. +lemma lfdeq_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 (lfxs_inv_zero … H) -H * /3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/ qed-. *) -lemma lfdeq_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}. +lemma lfdeq_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 lfxs_inv_lref/ qed-. (* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *) -lemma lfdeq_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. +lemma lfdeq_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 lfxs_inv_bind/ qed-. (* Basic_2A1: uses: lleq_inv_flat *) -lemma lfdeq_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. +lemma lfdeq_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 lfxs_inv_flat/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lfdeq_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. +lemma lfdeq_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 lfxs_inv_zero_pair_sn/ qed-. -lemma lfdeq_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. +lemma lfdeq_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 lfxs_inv_zero_pair_dx/ qed-. -lemma lfdeq_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}. +lemma lfdeq_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 lfxs_inv_lref_bind_sn/ qed-. -lemma lfdeq_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}. +lemma lfdeq_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 lfxs_inv_lref_bind_dx/ qed-. (* Basic forward lemmas *****************************************************) -lemma lfdeq_fwd_zero_pair: ∀h,o,I,K1,K2,V1,V2. - K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2. +lemma lfdeq_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 lfxs_fwd_zero_pair/ qed-. (* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *) -lemma lfdeq_fwd_pair_sn: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[h, o, V] L2. +lemma lfdeq_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 lfxs_fwd_pair_sn/ qed-. (* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *) -lemma lfdeq_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. +lemma lfdeq_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 lfxs_fwd_bind_dx/ qed-. (* Basic_2A1: uses: lleq_fwd_flat_dx *) -lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2. +lemma lfdeq_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 lfxs_fwd_flat_dx/ qed-. -lemma lfdeq_fwd_dx: ∀h,o,I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} → - ∃∃I1,K1. L1 = K1.ⓘ{I1}. +lemma lfdeq_fwd_dx (h) (o): ∀I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} → + ∃∃I1,K1. L1 = K1.ⓘ{I1}. /2 width=5 by lfxs_fwd_dx/ qed-. - -(* Basic_2A1: removed theorems 10: - lleq_ind lleq_fwd_lref - lleq_fwd_drop_sn lleq_fwd_drop_dx - lleq_skip lleq_lref lleq_free - lleq_Y lleq_ge_up lleq_ge - -*)