X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fldrop.ma;h=d209bd4b0f094b93a5e2109e6a084306e458562c;hb=f95f6cb21b86f3dad114b21f687aa5df36088064;hp=f51640fa13f69b220a7d83a97c94120d61454954;hpb=80178d6cf86b78bb9fc47f397f4bcfb1fd15a24f;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop.ma index f51640fa1..d209bd4b0 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "basic_2/notation/relations/rdrop_4.ma". include "basic_2/grammar/lenv_length.ma". include "basic_2/grammar/lenv_weight.ma". include "basic_2/relocation/lift.ma". @@ -20,7 +21,7 @@ include "basic_2/relocation/lift.ma". (* Basic_1: includes: drop_skip_bind *) inductive ldrop: nat → nat → relation lenv ≝ -| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆) +| ldrop_atom : ∀d. ldrop d 0 (⋆) (⋆) | ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V) | ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2 | ldrop_skip : ∀L1,L2,I,V1,V2,d,e. @@ -53,53 +54,40 @@ definition dropable_dx: predicate (relation lenv) ≝ (* Basic inversion lemmas ***************************************************) -fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2. -#d #e #L1 #L2 * -d -e -L1 -L2 -[ // -| // -| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct -] -qed. - -(* Basic_1: was: drop_gen_refl *) -lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2. -/2 width=5/ qed-. - fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ → - L2 = ⋆. + L2 = ⋆ ∧ e = 0. #d #e #L1 #L2 * -d -e -L1 -L2 -[ // +[ /2 width=1/ | #L #I #V #H destruct | #L1 #L2 #I #V #e #_ #H destruct | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct ] -qed. +qed-. (* Basic_1: was: drop_gen_sort *) -lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆. -/2 width=5/ qed-. +lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆ ∧ e = 0. +/2 width=4 by ldrop_inv_atom1_aux/ qed-. -fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → - ∀K,I,V. L1 = K. ⓑ{I} V → - (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ - (0 < e ∧ ⇩[d, e - 1] K ≡ L2). +fact ldrop_inv_O1_pair1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → + ∀K,I,V. L1 = K. ⓑ{I} V → + (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ + (0 < e ∧ ⇩[d, e - 1] K ≡ L2). #d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #K #I #V #H destruct +[ #d #_ #K #I #V #H destruct | #L #I #V #_ #K #J #W #HX destruct /3 width=1/ | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/ | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct ] -qed. +qed-. -lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 → - (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ - (0 < e ∧ ⇩[0, e - 1] K ≡ L2). -/2 width=3/ qed-. +lemma ldrop_inv_O1_pair1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 → + (e = 0 ∧ L2 = K. ⓑ{I} V) ∨ + (0 < e ∧ ⇩[0, e - 1] K ≡ L2). +/2 width=3 by ldrop_inv_O1_pair1_aux/ qed-. lemma ldrop_inv_pair1: ∀K,I,V,L2. ⇩[0, 0] K. ⓑ{I} V ≡ L2 → L2 = K. ⓑ{I} V. #K #I #V #L2 #H -elim (ldrop_inv_O1 … H) -H * // #H destruct +elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct elim (lt_refl_false … H) qed-. @@ -107,7 +95,7 @@ qed-. lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2. #e #K #I #V #L2 #H #He -elim (ldrop_inv_O1 … H) -H * // #H destruct +elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct elim (lt_refl_false … He) qed-. @@ -117,27 +105,27 @@ fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d → ⇧[d - 1, e] V2 ≡ V1 & L2 = K2. ⓑ{I} V2. #d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K #V #H destruct +[ #d #_ #I #K #V #H destruct | #L #I #V #H elim (lt_refl_false … H) | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) | #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/ ] -qed. +qed-. (* Basic_1: was: drop_gen_skip_l *) lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d → ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 & L2 = K2. ⓑ{I} V2. -/2 width=3/ qed-. +/2 width=3 by ldrop_inv_skip1_aux/ qed-. lemma ldrop_inv_O1_pair2: ∀I,K,V,e,L1. ⇩[0, e] L1 ≡ K. ⓑ{I} V → (e = 0 ∧ L1 = K. ⓑ{I} V) ∨ ∃∃I1,K1,V1. ⇩[0, e - 1] K1 ≡ K. ⓑ{I} V & L1 = K1.ⓑ{I1}V1 & 0 < e. #I #K #V #e * -[ #H lapply (ldrop_inv_atom1 … H) -H #H destruct +[ #H elim (ldrop_inv_atom1 … H) -H #H destruct | #L1 #I1 #V1 #H - elim (ldrop_inv_O1 … H) -H * + elim (ldrop_inv_O1_pair1 … H) -H * [ #H1 #H2 destruct /3 width=1/ | /3 width=5/ ] @@ -150,24 +138,25 @@ fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d → ⇧[d - 1, e] V2 ≡ V1 & L1 = K1. ⓑ{I} V1. #d #e #L1 #L2 * -d -e -L1 -L2 -[ #d #e #_ #I #K #V #H destruct +[ #d #_ #I #K #V #H destruct | #L #I #V #H elim (lt_refl_false … H) | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/ ] -qed. +qed-. (* Basic_1: was: drop_gen_skip_r *) lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d → ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 & L1 = K1. ⓑ{I} V1. -/2 width=3/ qed-. +/2 width=3 by ldrop_inv_skip2_aux/ qed-. (* Basic properties *********************************************************) (* Basic_1: was by definition: drop_refl *) -lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L. +lemma ldrop_refl: ∀L,d. ⇩[d, 0] L ≡ L. #L elim L -L // +#L #I #V #IHL #d @(nat_ind_plus … d) -d // /2 width=1/ qed. lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e. @@ -181,21 +170,21 @@ lemma ldrop_skip_lt: ∀L1,L2,I,V1,V2,d,e. #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/ qed. -lemma ldrop_O1_le: ∀i,L. i ≤ |L| → ∃K. ⇩[0, i] L ≡ K. -#i @(nat_ind_plus … i) -i /2 width=2/ -#i #IHi * +lemma ldrop_O1_le: ∀e,L. e ≤ |L| → ∃K. ⇩[0, e] L ≡ K. +#e @(nat_ind_plus … e) -e /2 width=2/ +#e #IHe * [ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct | #L #I #V normalize #H - elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/ + elim (IHe L) -IHe /2 width=1/ -H /3 width=2/ ] qed. -lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V. +lemma ldrop_O1_lt: ∀L,e. e < |L| → ∃∃I,K,V. ⇩[0, e] L ≡ K.ⓑ{I}V. #L elim L -L -[ #i #H elim (lt_zero_false … H) -| #L #I #V #IHL #i @(nat_ind_plus … i) -i /2 width=4/ - #i #_ normalize #H - elim (IHL i ? ) -IHL /2 width=1/ -H /3 width=4/ +[ #e #H elim (lt_zero_false … H) +| #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4/ + #e #_ normalize #H + elim (IHL e) -IHL /2 width=1/ -H /3 width=4/ ] qed. @@ -247,78 +236,91 @@ qed. (* Basic forvard lemmas *****************************************************) -lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize -[ /2 width=3/ -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 - >(lift_fwd_tw … HV21) -HV21 /2 width=1/ -] -qed-. - (* Basic_1: was: drop_S *) lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 → ⇩[O, e + 1] L1 ≡ K2. #L1 elim L1 -L1 -[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct +[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H * #H destruct | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #H + elim (ldrop_inv_O1_pair1 … H) -H * #He #H [ -IHL1 destruct /2 width=1/ | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/ ] ] qed-. -lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| ≤ |L1|. +lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L1| = |L2| + e. #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1/ qed-. -lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e. - ⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|. -#L1 elim L1 -L1 -[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct -| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #H - [ -IHL1 destruct // - | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/ - ] -] +lemma ldrop_fwd_length_minus2: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| = |L1| - e. +#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1/ qed-. -lemma ldrop_fwd_O1_length: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → |L2| = |L1| - e. -#L1 elim L1 -L1 -[ #L2 #e #H >(ldrop_inv_atom1 … H) -H // -| #K1 #I1 #V1 #IHL1 #L2 #e #H - elim (ldrop_inv_O1 … H) -H * #He #H - [ -IHL1 destruct // - | lapply (IHL1 … H) -IHL1 -H #H >H -H normalize - >minus_le_minus_minus_comm // - ] -] +lemma ldrop_fwd_length_minus4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → e = |L1| - |L2|. +#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H // qed-. -lemma ldrop_fwd_lw_eq: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → - |L1| = |L2| → ♯{L2} = ♯{L1}. -#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // -[ #L1 #L2 #I #V #e #HL12 #_ - lapply (ldrop_fwd_O1_length … HL12) -HL12 #HL21 >HL21 -HL21 normalize #H -I - lapply (discr_plus_xy_minus_xz … H) -e #H destruct -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #HL12 normalize in ⊢ (??%%→??%%); #H -I - >(lift_fwd_tw … HV21) -V2 /3 width=1 by eq_f2/ (**) (* auto is a bit slow without trace *) +lemma ldrop_fwd_length_le2: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → e ≤ |L1|. +#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H // +qed-. + +lemma ldrop_fwd_length_le4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| ≤ |L1|. +#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H // +qed-. + +lemma ldrop_fwd_length_lt2: ∀L1,I2,K2,V2,d,e. + ⇩[d, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|. +#L1 #I2 #K2 #V2 #d #e #H +lapply (ldrop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 // +qed-. + +lemma ldrop_fwd_length_lt4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → 0 < e → |L2| < |L1|. +#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1/ +qed-. + +lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}. +#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize +[ /2 width=3/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 + >(lift_fwd_tw … HV21) -HV21 /2 width=1/ ] qed-. -lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → - |L2| < |L1| → ♯{L2} < ♯{L1}. +lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → 0 < e → ♯{L2} < ♯{L1}. #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // [ #L #I #V #H elim (lt_refl_false … H) | #L1 #L2 #I #V #e #HL12 #_ #_ lapply (ldrop_fwd_lw … HL12) -HL12 #HL12 @(le_to_lt_to_lt … HL12) -HL12 // -| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 normalize in ⊢ (?%%→?%%); #H -I - >(lift_fwd_tw … HV21) -V2 /4 width=2 by lt_minus_to_plus, lt_plus_to_lt_l/ (**) (* auto too slow without trace *) +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I + >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/ (**) (* auto too slow without trace *) ] qed-. +(* Advanced inversion lemmas ************************************************) + +fact ldrop_inv_O2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → e = 0 → L1 = L2. +#d #e #L1 #L2 #H elim H -d -e -L1 -L2 +[ // +| // +| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 #H + >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -d -e // +] +qed-. + +(* Basic_1: was: drop_gen_refl *) +lemma ldrop_inv_O2: ∀L1,L2,d. ⇩[d, 0] L1 ≡ L2 → L1 = L2. +/2 width=4 by ldrop_inv_O2_aux/ qed-. + +lemma ldrop_inv_length_eq: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L1| = |L2| → e = 0. +#L1 #L2 #d #e #H #HL12 lapply (ldrop_fwd_length_minus4 … H) // +qed-. + +lemma ldrop_inv_refl: ∀L,d,e. ⇩[d, e] L ≡ L → e = 0. +/2 width=5 by ldrop_inv_length_eq/ qed-. + (* Basic_1: removed theorems 50: drop_ctail drop_skip_flat cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf