X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfdeq.ma;h=9784f34904ddc209745c628de2a3d92dee9894be;hb=a5c71699f1d0cf63a769c71dd8b8cd5dfff1933d;hp=f6bc891d7af225ef58240b517c592bdf69f83728;hpb=670ad7822d59e598a38d9037d482d3de188b170c;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 f6bc891d7..9784f3490 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma @@ -12,8 +12,8 @@ (* *) (**************************************************************************) -include "basic_2/notation/relations/lazyeq_5.ma". -include "basic_2/syntax/tdeq.ma". +include "basic_2/notation/relations/stareqsn_5.ma". +include "basic_2/syntax/tdeq_ext.ma". include "basic_2/static/lfxs.ma". (* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) @@ -23,165 +23,174 @@ definition lfdeq: ∀h. sd h → relation3 term lenv lenv ≝ interpretation "degree-based equivalence on referred entries (local environment)" - 'LazyEq h o T L1 L2 = (lfdeq h o T L1 L2). + 'StarEqSn h o T L1 L2 = (lfdeq h o T L1 L2). interpretation "degree-based ranged equivalence (local environment)" - 'LazyEq h o f L1 L2 = (lexs (cdeq 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. -*) + 'StarEqSn h o f L1 L2 = (lexs (cdeq_ext h o) cfull f L1 L2). + (* 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 #I1 #Hf #X #H1 elim (tdeq_fwd_atom1 … H1) -H1 - #I2 #H1 #Y #H2 lapply (lexs_inv_atom1 … H2) -H2 - #H2 destruct /2 width=1 by frees_atom/ -| #f #I #L1 #V1 #s1 #_ #IH #X #H1 elim (tdeq_inv_sort1 … H1) -H1 - #s2 #d #Hs1 #Hs2 #H1 #Y #H2 elim (lexs_inv_push1 … H2) -H2 - #L2 #V2 #HL12 #_ #H2 destruct /4 width=3 by frees_sort, tdeq_sort/ -| #f #I #L1 #V1 #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 - #Y #H2 elim (lexs_inv_next1 … H2) -H2 - #L2 #V2 #HL12 #HV12 #H2 destruct /3 width=1 by frees_zero/ -| #f #I #L1 #V1 #i #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 - #Y #H2 elim (lexs_inv_push1 … H2) -H2 - #L2 #V2 #HL12 #_ #H2 destruct /3 width=1 by frees_lref/ -| #f #I #L1 #V1 #l #_ #IH #X #H1 >(tdeq_inv_gref1 … H1) -H1 - #Y #H2 elim (lexs_inv_push1 … H2) -H2 - #L2 #V2 #HL12 #_ #H2 destruct /3 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, lexs_inv_tl, sle_lexs_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 +[ #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 + >(lexs_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 (lexs_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 (lexs_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 (lexs_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, lexs_inv_tl, ext2_pair, sle_lexs_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_lexs_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. -/3 width=7 by frees_tdeq_conf_lexs, lexs_refl/ qed-. + ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f. +/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-. -lemma frees_lfdeq_conf_lexs: ∀h,o. lexs_frees_confluent (cdeq h o) cfull. -/3 width=7 by frees_tdeq_conf_lexs, ex2_intro/ 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 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-. -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 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}] ⋆. +lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆. /2 width=1 by lfxs_atom/ qed. -lemma lfdeq_sort: ∀h,o,I,L1,L2,V1,V2,s. - L1 ≡[h, o, ⋆s] L2 → L1.ⓑ{I}V1 ≡[h, o, ⋆s] L2.ⓑ{I}V2. +(* 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}. /2 width=1 by lfxs_sort/ qed. -lemma lfdeq_zero: ∀h,o,I,L1,L2,V. - L1 ≡[h, o, V] L2 → L1.ⓑ{I}V ≡[h, o, #0] L2.ⓑ{I}V. -/2 width=1 by lfxs_zero/ qed. - -lemma lfdeq_lref: ∀h,o,I,L1,L2,V1,V2,i. - L1 ≡[h, o, #i] L2 → L1.ⓑ{I}V1 ≡[h, o, #⫯i] 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}. +/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}. /2 width=1 by lfxs_lref/ qed. -lemma lfdeq_gref: ∀h,o,I,L1,L2,V1,V2,l. - L1 ≡[h, o, §l] L2 → L1.ⓑ{I}V1 ≡[h, o, §l] L2.ⓑ{I}V2. +(* 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}. /2 width=1 by lfxs_gref/ qed. -lemma lfdeq_pair_repl_dx: ∀h,o,I,L1,L2.∀T:term.∀V,V1. - L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V1 → - ∀V2. V ≡[h, o] V2 → - L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V2. -/2 width=2 by lfxs_pair_repl_dx/ 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}. +/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. +(* +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 ex4_5_intro, or_introl, or_intror, conj/ +/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 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ≡[h, o, #i] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +*) +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-. -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. +(* 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. /2 width=2 by lfxs_inv_bind/ qed-. -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. +(* 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. /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. -#h #o #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=5 by ex3_2_intro/ -qed-. +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. -#h #o #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero_pair_dx … H) -H -#L1 #V1 #HL12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ -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. +/2 width=1 by lfxs_inv_zero_pair_dx/ qed-. -lemma lfdeq_inv_lref_pair_sn: ∀h,o,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[h, o, #⫯i] Y2 → - ∃∃L2,V2. L1 ≡[h, o, #i] L2 & Y2 = L2.ⓑ{I}V2. -/2 width=2 by lfxs_inv_lref_pair_sn/ 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}. +/2 width=2 by lfxs_inv_lref_bind_sn/ qed-. -lemma lfdeq_inv_lref_pair_dx: ∀h,o,I,Y1,L2,V2,i. Y1 ≡[h, o, #⫯i] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ≡[h, o, #i] L2 & Y1 = L1.ⓑ{I}V1. -/2 width=2 by lfxs_inv_lref_pair_dx/ 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}. +/2 width=2 by lfxs_inv_lref_bind_dx/ qed-. (* Basic forward lemmas *****************************************************) -lemma lfdeq_fwd_bind_sn: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1 ≡[h, o, V] L2. -/2 width=4 by lfxs_fwd_bind_sn/ qed-. +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. +/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. + 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-. -lemma lfdeq_fwd_flat_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, V] L2. -/2 width=3 by lfxs_fwd_flat_sn/ qed-. - -lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, T] L2. +(* 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. /2 width=3 by lfxs_fwd_flat_dx/ qed-. -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-. - -lemma lfdeq_fwd_dx: ∀h,o,I,L1,K2,V2. ∀T:term. L1 ≡[h, o, T] K2.ⓑ{I}V2 → - ∃∃K1,V1. L1 = K1.ⓑ{I}V1. +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 31: - lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref - lleq_fwd_drop_sn lleq_fwd_drop_dx - lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx - lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat - lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl - lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div - lleq_dec -*)