X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fmultiple%2Fdrops.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fmultiple%2Fdrops.ma;h=a86afce84bf3776e8a25a182d69e27eb0b649168;hb=d2545ffd201b1aa49887313791386add78fa8603;hp=0000000000000000000000000000000000000000;hpb=57ae1762497a5f3ea75740e2908e04adb8642cc2;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma new file mode 100644 index 000000000..a86afce84 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma @@ -0,0 +1,122 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2A/notation/relations/rdropstar_3.ma". +include "basic_2A/notation/relations/rdropstar_4.ma". +include "basic_2A/substitution/drop.ma". +include "basic_2A/multiple/mr2_minus.ma". +include "basic_2A/multiple/lifts_vector.ma". + +(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) + +inductive drops (s:bool): list2 nat nat → relation lenv ≝ +| drops_nil : ∀L. drops s (◊) L L +| drops_cons: ∀L1,L,L2,cs,l,m. + drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s ({l, m} @ cs) L1 L2 +. + +interpretation "iterated slicing (local environment) abstract" + 'RDropStar s cs T1 T2 = (drops s cs T1 T2). +(* +interpretation "iterated slicing (local environment) general" + 'RDropStar des T1 T2 = (drops true des T1 T2). +*) + +definition d_liftable1: relation2 lenv term → predicate bool ≝ + λR,s. ∀K,T. R K T → ∀L,l,m. ⬇[s, l, m] L ≡ K → + ∀U. ⬆[l, m] T ≡ U → R L U. + +definition d_liftables1: relation2 lenv term → predicate bool ≝ + λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K → + ∀T,U. ⬆*[cs] T ≡ U → R K T → R L U. + +definition d_liftables1_all: relation2 lenv term → predicate bool ≝ + λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K → + ∀Ts,Us. ⬆*[cs] Ts ≡ Us → + all … (R K) Ts → all … (R L) Us. + +(* Basic inversion lemmas ***************************************************) + +fact drops_inv_nil_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → cs = ◊ → L1 = L2. +#L1 #L2 #s #cs * -L1 -L2 -cs // +#L1 #L #L2 #l #m #cs #_ #_ #H destruct +qed-. + +(* Basic_1: was: drop1_gen_pnil *) +lemma drops_inv_nil: ∀L1,L2,s. ⬇*[s, ◊] L1 ≡ L2 → L1 = L2. +/2 width=4 by drops_inv_nil_aux/ qed-. + +fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → + ∀l,m,tl. cs = {l, m} @ tl → + ∃∃L. ⬇*[s, tl] L1 ≡ L & ⬇[s, l, m] L ≡ L2. +#L1 #L2 #s #cs * -L1 -L2 -cs +[ #L #l #m #tl #H destruct +| #L1 #L #L2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct + /2 width=3 by ex2_intro/ +] +qed-. + +(* Basic_1: was: drop1_gen_pcons *) +lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 → + ∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2. +/2 width=3 by drops_inv_cons_aux/ qed-. + +lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≡ cs2 → + ∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 → + ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≡ cs1 + 1 & + ⬇*[s, cs1] K1 ≡ K2 & + ⬆*[cs1] V2 ≡ V1 & + L1 = K1. ⓑ{I} V1. +#I #s #cs #cs2 #i #H elim H -cs -cs2 -i +[ #i #L1 #K2 #V2 #H + >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/ +| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H + elim (drops_inv_cons … H) -H #L #HL1 #H + elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct + elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct + @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ] + normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *) +| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H + elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct + /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/ +] +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: drop1_skip_bind *) +lemma drops_skip: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → ∀V1,V2. ⬆*[cs] V2 ≡ V1 → + ∀I. ⬇*[s, cs + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2. +#L1 #L2 #s #cs #H elim H -L1 -L2 -cs +[ #L #V1 #V2 #HV12 #I + >(lifts_inv_nil … HV12) -HV12 // +| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #V1 #V2 #H #I + elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/ +]. +qed. + +lemma d1_liftable_liftables: ∀R,s. d_liftable1 R s → d_liftables1 R s. +#R #s #HR #L #K #cs #H elim H -L -K -cs +[ #L #T #U #H #HT <(lifts_inv_nil … H) -H // +| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2 + elim (lifts_inv_cons … H) -H /3 width=10 by/ +] +qed. + +lemma d1_liftables_liftables_all: ∀R,s. d_liftables1 R s → d_liftables1_all R s. +#R #s #HR #L #K #cs #HLK #Ts #Us #H elim H -Ts -Us normalize // +#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/ +qed. + +(* Basic_1: removed theorems 1: drop1_getl_trans *)