X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfeq.ma;h=f7375afbc829971da4837150469b46705fab7f0d;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=28da143b8c2b18fffda39799ec1c0e684705e60c;hpb=e39d1924cd572acdf0cf8dba08f3b650dfd6abee;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma index 28da143b8..f7375afbc 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma @@ -12,125 +12,98 @@ (* *) (**************************************************************************) -include "basic_2/notation/relations/lazyeq_3.ma". +include "basic_2/notation/relations/ideqsn_3.ma". include "basic_2/static/lfxs.ma". -(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************) +(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********) -definition lfeq: relation3 term lenv lenv ≝ lfxs ceq. +(* Basic_2A1: was: lleq *) +definition lfeq: relation3 term lenv lenv ≝ + lfxs ceq. interpretation - "equivalence on referred entries (local environment)" - 'LazyEq T L1 L2 = (lfeq T L1 L2). + "syntactic equivalence on referred entries (local environment)" + 'IdEqSn T L1 L2 = (lfeq T L1 L2). +(* Note: "lfeq_transitive R" is equivalent to "lfxs_transitive ceq R R" *) +(* Basic_2A1: uses: lleq_transitive *) definition lfeq_transitive: predicate (relation3 lenv term term) ≝ λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2. -(* Basic properties ***********************************************************) - -lemma lfeq_atom: ∀I. ⋆ ≡[⓪{I}] ⋆. -/2 width=1 by lfxs_atom/ qed. - -lemma lfeq_sort: ∀I,L1,L2,V1,V2,s. - L1 ≡[⋆s] L2 → L1.ⓑ{I}V1 ≡[⋆s] L2.ⓑ{I}V2. -/2 width=1 by lfxs_sort/ qed. - -lemma lfeq_zero: ∀I,L1,L2,V. - L1 ≡[V] L2 → L1.ⓑ{I}V ≡[#0] L2.ⓑ{I}V. -/2 width=1 by lfxs_zero/ qed. - -lemma lfeq_lref: ∀I,L1,L2,V1,V2,i. - L1 ≡[#i] L2 → L1.ⓑ{I}V1 ≡[#⫯i] L2.ⓑ{I}V2. -/2 width=1 by lfxs_lref/ qed. - -lemma lfeq_gref: ∀I,L1,L2,V1,V2,l. - L1 ≡[§l] L2 → L1.ⓑ{I}V1 ≡[§l] L2.ⓑ{I}V2. -/2 width=1 by lfxs_gref/ qed. - (* Basic inversion lemmas ***************************************************) -lemma lfeq_inv_atom_sn: ∀I,Y2. ⋆ ≡[⓪{I}] Y2 → Y2 = ⋆. -/2 width=3 by lfxs_inv_atom_sn/ qed-. - -lemma lfeq_inv_atom_dx: ∀I,Y1. Y1 ≡[⓪{I}] ⋆ → Y1 = ⋆. -/2 width=3 by lfxs_inv_atom_dx/ qed-. - -lemma lfeq_inv_zero: ∀Y1,Y2. Y1 ≡[#0] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V. L1 ≡[V] L2 & - Y1 = L1.ⓑ{I}V & Y2 = L2.ⓑ{I}V. -#Y1 #Y2 #H elim (lfxs_inv_zero … H) -H * -/3 width=7 by ex3_4_intro, or_introl, or_intror, conj/ -qed-. - -lemma lfeq_inv_lref: ∀Y1,Y2,i. Y1 ≡[#⫯i] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ≡[#i] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -#Y1 #Y2 #i #H elim (lfxs_inv_lref … H) -H * -/3 width=8 by ex3_5_intro, or_introl, or_intror, conj/ -qed-. - -lemma lfeq_inv_bind: ∀I,L1,L2,V,T,p. L1 ≡[ⓑ{p,I}V.T] L2 → - L1 ≡[V] L2 ∧ L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V. -#I #L1 #L2 #V #T #p #H elim (lfxs_inv_bind … H) -H /2 width=3 by conj/ -qed-. +lemma lfeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 → + ∧∧ L1 ≡[V] L2 & L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V. +/2 width=2 by lfxs_inv_bind/ qed-. lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → - L1 ≡[V] L2 ∧ L1 ≡[T] L2. -#I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H /2 width=3 by conj/ -qed-. + ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2. +/2 width=2 by lfxs_inv_flat/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lfeq_inv_zero_pair_sn: ∀I,Y2,L1,V. L1.ⓑ{I}V ≡[#0] Y2 → - ∃∃L2. L1 ≡[V] L2 & Y2 = L2.ⓑ{I}V. -#I #Y2 #L1 #V #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=3 by ex2_intro/ +lemma lfeq_inv_zero_pair_sn: ∀I,L2,K1,V. K1.ⓑ{I}V ≡[#0] L2 → + ∃∃K2. K1 ≡[V] K2 & L2 = K2.ⓑ{I}V. +#I #L2 #K1 #V #H +elim (lfxs_inv_zero_pair_sn … H) -H #K2 #X #HK12 #HX #H destruct +/2 width=3 by ex2_intro/ qed-. -lemma lfeq_inv_zero_pair_dx: ∀I,Y1,L2,V. Y1 ≡[#0] L2.ⓑ{I}V → - ∃∃L1. L1 ≡[V] L2 & Y1 = L1.ⓑ{I}V. -#I #Y1 #L2 #V #H elim (lfxs_inv_zero_pair_dx … H) -H -#L1 #X #HL12 #HX #H destruct /2 width=3 by ex2_intro/ +lemma lfeq_inv_zero_pair_dx: ∀I,L1,K2,V. L1 ≡[#0] K2.ⓑ{I}V → + ∃∃K1. K1 ≡[V] K2 & L1 = K1.ⓑ{I}V. +#I #L1 #K2 #V #H +elim (lfxs_inv_zero_pair_dx … H) -H #K1 #X #HK12 #HX #H destruct +/2 width=3 by ex2_intro/ qed-. -lemma lfeq_inv_lref_pair_sn: ∀I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[#⫯i] Y2 → - ∃∃L2,V2. L1 ≡[#i] L2 & Y2 = L2.ⓑ{I}V2. -/2 width=2 by lfxs_inv_lref_pair_sn/ qed-. +lemma lfeq_inv_lref_bind_sn: ∀I1,K1,L2,i. K1.ⓘ{I1} ≡[#↑i] L2 → + ∃∃I2,K2. K1 ≡[#i] K2 & L2 = K2.ⓘ{I2}. +/2 width=2 by lfxs_inv_lref_bind_sn/ qed-. -lemma lfeq_inv_lref_pair_dx: ∀I,Y1,L2,V2,i. Y1 ≡[#⫯i] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ≡[#i] L2 & Y1 = L1.ⓑ{I}V1. -/2 width=2 by lfxs_inv_lref_pair_dx/ qed-. +lemma lfeq_inv_lref_bind_dx: ∀I2,K2,L1,i. L1 ≡[#↑i] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ{I1}. +/2 width=2 by lfxs_inv_lref_bind_dx/ qed-. (* Basic forward lemmas *****************************************************) -lemma lfeq_fwd_bind_sn: ∀I,L1,L2,V,T,p. L1 ≡[ⓑ{p,I}V.T] L2 → L1 ≡[V] L2. -/2 width=4 by lfxs_fwd_bind_sn/ qed-. - -lemma lfeq_fwd_bind_dx: ∀I,L1,L2,V,T,p. - L1 ≡[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V. -/2 width=2 by lfxs_fwd_bind_dx/ qed-. - -lemma lfeq_fwd_flat_sn: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[V] L2. -/2 width=3 by lfxs_fwd_flat_sn/ qed-. - -lemma lfeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[T] L2. -/2 width=3 by lfxs_fwd_flat_dx/ qed-. - -lemma lfeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≡[②{I}V.T] L2 → L1 ≡[V] L2. -/2 width=3 by lfxs_fwd_pair_sn/ qed-. - -(* Advanceded forward lemmas with generic extension on referred entries *****) +(* Basic_2A1: was: llpx_sn_lrefl *) +(* Note: this should have been lleq_fwd_llpx_sn *) +lemma lfeq_fwd_lfxs: ∀R. c_reflexive … R → + ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤*[R, T] L2. +#R #HR #L1 #L2 #T * #f #Hf #HL12 +/4 width=7 by lexs_co, cext2_co, ex2_intro/ +qed-. -lemma lfex_fwd_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → - ∀L1,L2,T. L1 ≡[T] L2 → L1 ⦻*[R, T] L2. -/2 width=3 by lfxs_co/ qed-. +(* Basic_properties *********************************************************) + +lemma frees_lfeq_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f → + ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f. +#f #L1 #T #H elim H -f -L1 -T +[ /2 width=3 by frees_sort/ +| #f #i #Hf #L2 #H2 + >(lfxs_inv_atom_sn … H2) -L2 + /2 width=1 by frees_atom/ +| #f #I #L1 #V1 #_ #IH #Y #H2 + elim (lfeq_inv_zero_pair_sn … H2) -H2 #L2 #HL12 #H destruct + /3 width=1 by frees_pair/ +| #f #I #L1 #Hf #Y #H2 + elim (lfxs_inv_zero_unit_sn … H2) -H2 #g #L2 #_ #_ #H destruct + /2 width=1 by frees_unit/ +| #f #I #L1 #i #_ #IH #Y #H2 + elim (lfeq_inv_lref_bind_sn … H2) -H2 #J #L2 #HL12 #H destruct + /3 width=1 by frees_lref/ +| /2 width=1 by frees_gref/ +| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2 + elim (lfeq_inv_bind … H2) -H2 /3 width=5 by frees_bind/ +| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2 + elim (lfeq_inv_flat … H2) -H2 /3 width=5 by frees_flat/ +] +qed-. -(* Basic_2A1: removed theorems 30: - lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref +(* Basic_2A1: removed theorems 10: + lleq_ind 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_skip lleq_lref lleq_free + lleq_Y lleq_ge_up lleq_ge + *)