X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Flleq_ldrop.ma;h=27b16110deb53543994b092d2b98052416f5c0fc;hb=1555848a5546d0154964286d3400114481d78962;hp=9cb597c6f7be12cff7ac7a055962fd37bb424fd0;hpb=e5378812c068074f0ac627541d3f066e135c8824;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma index 9cb597c6f..27b16110d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma @@ -12,112 +12,139 @@ (* *) (**************************************************************************) -include "basic_2/substitution/cpys_lift.ma". +include "basic_2/relocation/llpx_sn_ldrop.ma". include "basic_2/substitution/lleq.ma". (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) (* Advanced properties ******************************************************) -lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ⋕[#i, d] L2. -#L1 #L2 #d #i #Hid #HL12 @conj // -HL12 -#U @conj #H elim (cpys_inv_lref1 … H) -H // * -#I #Z #Y #X #H elim (ylt_yle_false … Hid … H) -qed. - -lemma lleq_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i → - ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V → - K1 ⋕[V, 0] K2 → L1 ⋕[#i, d] L2. -#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ] -[ lapply (ldrop_fwd_length … HLK1) -HLK1 #H1 - lapply (ldrop_fwd_length … HLK2) -HLK2 #H2 - >H1 >H2 -H1 -H2 normalize // -| #U @conj #H elim (cpys_inv_lref1 … H) -H // * - >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU - [ letin HLK ≝ HLK1 | letin HLK ≝ HLK2 ] - lapply (ldrop_mono … H … HLK) -H #H destruct elim (IH W) - /3 width=7 by cpys_subst_Y2/ +lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V → + ∀J,W. L1 ⋕[W, 0] L2 → L1.ⓑ{J}W ⋕[T, 0] L2.ⓑ{J}W. +/2 width=7 by llpx_sn_bind_repl_O/ qed-. + +lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ⋕[T, d] L2). +/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-. + +lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R → + ∀L1,L2,T,d. L1 ⋕[T, d] L2 → + ∀L. llpx_sn R d T L2 L → llpx_sn R d T L1 L. +#R #HR #L1 #L2 #T #d #H @(lleq_ind … H) -L1 -L2 -T -d +[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/ +|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/ +| #I #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2 + /3 width=11 by llpx_sn_lref/ +| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H + /3 width=1 by llpx_sn_bind/ +| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H + /3 width=1 by llpx_sn_flat/ ] -qed. +qed-. + +lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R → + ∀L1,L2,T,d. L1 ⋕[T, d] L2 → + ∀L. llpx_sn R d T L1 L → llpx_sn R d T L2 L. +/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lleq_inv_lref_ge_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i → + ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V → + ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ⋕[V, 0] K2. +#L1 #L2 #d #i #H #Hdi #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2 +/2 width=3 by ex2_intro/ +qed-. + +lemma lleq_inv_lref_ge_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i → + ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → + ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ⋕[V, 0] K2. +#L1 #L2 #d #i #H #Hdi #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1 +/2 width=3 by ex2_intro/ +qed-. + +lemma lleq_inv_lref_ge_bi: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i → + ∀I1,I2,K1,K2,V1,V2. + ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & K1 ⋕[V1, 0] K2 & V1 = V2. +/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. + +lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i → + ∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V → + K1 ⋕[V, 0] K2. +#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2 +elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) // +qed-. + +lemma lleq_inv_S: ∀L1,L2,T,d. L1 ⋕[T, d+1] L2 → + ∀I,K1,K2,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V → + K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2. +/2 width=9 by llpx_sn_inv_S/ qed-. + +lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 → + L1 ⋕[V, 0] L2 ∧ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_inv_bind_O/ qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → + ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V → + i < d ∨ + ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ⋕[V, 0] K2 & d ≤ i. +#L1 #L2 #d #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 +[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ +qed-. -lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ⋕[#i, d] L2. -#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12 -#U @conj #H elim (cpys_inv_lref1 … H) -H // * -#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 … H) -H -#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/ -qed. +lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → + ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → + i < d ∨ + ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ⋕[V, 0] K2 & d ≤ i. +#L1 #L2 #d #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 +[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ +qed-. + +lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 → + L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-. (* Properties on relocation *************************************************) lemma lleq_lift_le: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 → ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 → ∀U. ⇧[d, e] T ≡ U → dt ≤ d → L1 ⋕[U, dt] L2. -#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd -lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2) -#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0 -[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ] -elim (cpys_inv_lift1_be … HU0 … HLKA … HTU) // -HU0 >yminus_Y_inj #T0 #HT0 #HTU0 -elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=12 by cpys_lift_be/ -qed-. +/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-. lemma lleq_lift_ge: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 → ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 → ∀U. ⇧[d, e] T ≡ U → d ≤ dt → L1 ⋕[U, dt+e] L2. -#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt -lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2) -#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0 -[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ] -elim (cpys_inv_lift1_ge … HU0 … HLKA … HTU) /2 width=1 by monotonic_yle_plus_dx/ -HU0 >yplus_minus_inj #T0 #HT0 #HTU0 -elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=10 by cpys_lift_ge/ -qed-. +/2 width=9 by llpx_sn_lift_ge/ qed-. (* Inversion lemmas on relocation *******************************************) lemma lleq_inv_lift_le: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 → ∀T. ⇧[d, e] T ≡ U → dt ≤ d → K1 ⋕[T, dt] K2. -#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd -lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2) -#H2 #H1 @conj // -HL12 -H1 -H2 -#T0 elim (lift_total T0 d e) -#U0 #HTU0 elim (IH U0) -IH -#H12 #H21 @conj #HT0 -[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ] -lapply (cpys_lift_be … HT0 … HLKA … HTU … HTU0) // -HT0 ->yplus_Y1 #HU0 elim (cpys_inv_lift1_be … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdtd -#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 // -qed-. +/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-. + +lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → + ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 → + ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ⋕[T, d] K2. +/2 width=11 by llpx_sn_inv_lift_be/ qed-. lemma lleq_inv_lift_ge: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 → ∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ dt → K1 ⋕[T, dt-e] K2. -#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt -lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2) -#H2 #H1 @conj // -HL12 -H1 -H2 -elim (yle_inv_plus_inj2 … Hdedt) #Hddt #Hedt -#T0 elim (lift_total T0 d e) -#U0 #HTU0 elim (IH U0) -IH -#H12 #H21 @conj #HT0 -[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ] -lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0 -Hddt ->ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdedt -Hedt -#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 // -qed-. +/2 width=9 by llpx_sn_inv_lift_ge/ qed-. -lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → - ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 → - ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ⋕[T, d] K2. -#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde -lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2) -#H2 #H1 @conj // -HL12 -H1 -H2 -#T0 elim (lift_total T0 d e) -#U0 #HTU0 elim (IH U0) -IH -#H12 #H21 @conj #HT0 -[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ] -lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0 -#HU0 lapply (cpys_weak … HU0 dt (∞) ? ?) // -HU0 -#HU0 lapply (H0 HU0) -#HU0 lapply (cpys_weak … HU0 d (∞) ? ?) // -HU0 -#HU0 elim (cpys_inv_lift1_ge_up … HU0 … HLKB … HTU) // -L1 -L2 -U -Hddt -Hdtde -#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 // -qed-. +(* Inversion lemmas on negated lazy quivalence for local environments *******) + +lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,d. (L1 ⋕[ⓑ{a,I}V.T, d] L2 → ⊥) → + (L1 ⋕[V, d] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V → ⊥). +/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-. + +lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ⋕[ⓕ{I}V.T, d] L2 → ⊥) → + (L1 ⋕[V, d] L2 → ⊥) ∨ (L1 ⋕[T, d] L2 → ⊥). +/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-. + +lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ⋕[ⓑ{a,I}V.T, 0] L2 → ⊥) → + (L1 ⋕[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V → ⊥). +/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.