X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flifts.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flifts.ma;h=0ca46fa3c40120ad5e70fc5917055f1f4205c5cf;hb=7593c0f74b944fb100493fb24b665ce3b8d1d252;hp=0000000000000000000000000000000000000000;hpb=361a91ade954f92013da892c62d41e3a7168cfc0;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma new file mode 100644 index 000000000..0ca46fa3c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma @@ -0,0 +1,348 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "ground_2/relocation/trace_isid.ma". +include "basic_2/notation/relations/rliftstar_3.ma". +include "basic_2/grammar/term.ma". + +(* GENERIC TERM RELOCATION **************************************************) + +(* Basic_1: includes: + lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat + lifts_nil lifts_cons +*) +inductive lifts: trace → relation term ≝ +| lifts_sort: ∀k,t. lifts t (⋆k) (⋆k) +| lifts_lref: ∀i1,i2,t. @⦃i1, t⦄ ≡ i2 → lifts t (#i1) (#i2) +| lifts_gref: ∀p,t. lifts t (§p) (§p) +| lifts_bind: ∀a,I,V1,V2,T1,T2,t. + lifts t V1 V2 → lifts (Ⓣ@t) T1 T2 → + lifts t (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) +| lifts_flat: ∀I,V1,V2,T1,T2,t. + lifts t V1 V2 → lifts t T1 T2 → + lifts t (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) +. + +interpretation "generic relocation (term)" + 'RLiftStar cs T1 T2 = (lifts cs T1 T2). + +(* Basic inversion lemmas ***************************************************) + +fact lifts_inv_sort1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀k. X = ⋆k → Y = ⋆k. +#X #Y #t * -X -Y -t // +[ #i1 #i2 #t #_ #x #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +] +qed-. + +(* Basic_1: was: lift1_sort *) +(* Basic_2A1: includes: lift_inv_sort1 *) +lemma lifts_inv_sort1: ∀Y,k,t. ⬆*[t] ⋆k ≡ Y → Y = ⋆k. +/2 width=4 by lifts_inv_sort1_aux/ qed-. + +fact lifts_inv_lref1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀i1. X = #i1 → + ∃∃i2. @⦃i1, t⦄ ≡ i2 & Y = #i2. +#X #Y #t * -X -Y -t +[ #k #t #x #H destruct +| #i1 #i2 #t #Hi12 #x #H destruct /2 width=3 by ex2_intro/ +| #p #t #x #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +] +qed-. + +(* Basic_1: was: lift1_lref *) +(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *) +lemma lifts_inv_lref1: ∀Y,i1,t. ⬆*[t] #i1 ≡ Y → + ∃∃i2. @⦃i1, t⦄ ≡ i2 & Y = #i2. +/2 width=3 by lifts_inv_lref1_aux/ qed-. + +fact lifts_inv_gref1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀p. X = §p → Y = §p. +#X #Y #t * -X -Y -t // +[ #i1 #i2 #t #_ #x #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +] +qed-. + +(* Basic_2A1: includes: lift_inv_gref1 *) +lemma lifts_inv_gref1: ∀Y,p,t. ⬆*[t] §p ≡ Y → Y = §p. +/2 width=4 by lifts_inv_gref1_aux/ qed-. + +fact lifts_inv_bind1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → + ∀a,I,V1,T1. X = ⓑ{a,I}V1.T1 → + ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 & + Y = ⓑ{a,I}V2.T2. +#X #Y #t * -X -Y -t +[ #k #t #b #J #W1 #U1 #H destruct +| #i1 #i2 #t #_ #b #J #W1 #U1 #H destruct +| #p #t #b #J #W1 #U1 #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ +| #I #V1 #V2 #T1 #T2 #t #_ #_ #b #J #W1 #U1 #H destruct +] +qed-. + +(* Basic_1: was: lift1_bind *) +(* Basic_2A1: includes: lift_inv_bind1 *) +lemma lifts_inv_bind1: ∀a,I,V1,T1,Y,t. ⬆*[t] ⓑ{a,I}V1.T1 ≡ Y → + ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 & + Y = ⓑ{a,I}V2.T2. +/2 width=3 by lifts_inv_bind1_aux/ qed-. + +fact lifts_inv_flat1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → + ∀I,V1,T1. X = ⓕ{I}V1.T1 → + ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 & + Y = ⓕ{I}V2.T2. +#X #Y #t * -X -Y -t +[ #k #t #J #W1 #U1 #H destruct +| #i1 #i2 #t #_ #J #W1 #U1 #H destruct +| #p #t #J #W1 #U1 #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #J #W1 #U1 #H destruct +| #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +(* Basic_1: was: lift1_flat *) +(* Basic_2A1: includes: lift_inv_flat1 *) +lemma lifts_inv_flat1: ∀I,V1,T1,Y,t. ⬆*[t] ⓕ{I}V1.T1 ≡ Y → + ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 & + Y = ⓕ{I}V2.T2. +/2 width=3 by lifts_inv_flat1_aux/ qed-. + +fact lifts_inv_sort2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀k. Y = ⋆k → X = ⋆k. +#X #Y #t * -X -Y -t // +[ #i1 #i2 #t #_ #x #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +] +qed-. + +(* Basic_1: includes: lift_gen_sort *) +(* Basic_2A1: includes: lift_inv_sort2 *) +lemma lifts_inv_sort2: ∀X,k,t. ⬆*[t] X ≡ ⋆k → X = ⋆k. +/2 width=4 by lifts_inv_sort2_aux/ qed-. + +fact lifts_inv_lref2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀i2. Y = #i2 → + ∃∃i1. @⦃i1, t⦄ ≡ i2 & X = #i1. +#X #Y #t * -X -Y -t +[ #k #t #x #H destruct +| #i1 #i2 #t #Hi12 #x #H destruct /2 width=3 by ex2_intro/ +| #p #t #x #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +] +qed-. + +(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *) +(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *) +lemma lifts_inv_lref2: ∀X,i2,t. ⬆*[t] X ≡ #i2 → + ∃∃i1. @⦃i1, t⦄ ≡ i2 & X = #i1. +/2 width=3 by lifts_inv_lref2_aux/ qed-. + +fact lifts_inv_gref2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀p. Y = §p → X = §p. +#X #Y #t * -X -Y -t // +[ #i1 #i2 #t #_ #x #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct +] +qed-. + +(* Basic_2A1: includes: lift_inv_gref1 *) +lemma lifts_inv_gref2: ∀X,p,t. ⬆*[t] X ≡ §p → X = §p. +/2 width=4 by lifts_inv_gref2_aux/ qed-. + +fact lifts_inv_bind2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → + ∀a,I,V2,T2. Y = ⓑ{a,I}V2.T2 → + ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 & + X = ⓑ{a,I}V1.T1. +#X #Y #t * -X -Y -t +[ #k #t #b #J #W2 #U2 #H destruct +| #i1 #i2 #t #_ #b #J #W2 #U2 #H destruct +| #p #t #b #J #W2 #U2 #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #b #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/ +| #I #V1 #V2 #T1 #T2 #t #_ #_ #b #J #W2 #U2 #H destruct +] +qed-. + +(* Basic_1: includes: lift_gen_bind *) +(* Basic_2A1: includes: lift_inv_bind2 *) +lemma lifts_inv_bind2: ∀a,I,V2,T2,X,t. ⬆*[t] X ≡ ⓑ{a,I}V2.T2 → + ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 & + X = ⓑ{a,I}V1.T1. +/2 width=3 by lifts_inv_bind2_aux/ qed-. + +fact lifts_inv_flat2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → + ∀I,V2,T2. Y = ⓕ{I}V2.T2 → + ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 & + X = ⓕ{I}V1.T1. +#X #Y #t * -X -Y -t +[ #k #t #J #W2 #U2 #H destruct +| #i1 #i2 #t #_ #J #W2 #U2 #H destruct +| #p #t #J #W2 #U2 #H destruct +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #J #W2 #U2 #H destruct +| #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +(* Basic_1: includes: lift_gen_flat *) +(* Basic_2A1: includes: lift_inv_flat2 *) +lemma lifts_inv_flat2: ∀I,V2,T2,X,t. ⬆*[t] X ≡ ⓕ{I}V2.T2 → + ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 & + X = ⓕ{I}V1.T1. +/2 width=3 by lifts_inv_flat2_aux/ qed-. + +(* Basic_2A1: includes: lift_inv_pair_xy_x *) +lemma lifts_inv_pair_xy_x: ∀I,V,T,t. ⬆*[t] ②{I}V.T ≡ V → ⊥. +#J #V elim V -V +[ * #i #U #t #H + [ lapply (lifts_inv_sort2 … H) -H #H destruct + | elim (lifts_inv_lref2 … H) -H + #x #_ #H destruct + | lapply (lifts_inv_gref2 … H) -H #H destruct + ] +| * [ #a ] #I #V2 #T2 #IHV2 #_ #U #t #H + [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/ + | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/ + ] +] +qed-. + +(* Basic_1: includes: thead_x_lift_y_y *) +(* Basic_2A1: includes: lift_inv_pair_xy_y *) +lemma lifts_inv_pair_xy_y: ∀I,T,V,t. ⬆*[t] ②{I}V.T ≡ T → ⊥. +#J #T elim T -T +[ * #i #W #t #H + [ lapply (lifts_inv_sort2 … H) -H #H destruct + | elim (lifts_inv_lref2 … H) -H + #x #_ #H destruct + | lapply (lifts_inv_gref2 … H) -H #H destruct + ] +| * [ #a ] #I #V2 #T2 #_ #IHT2 #W #t #H + [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/ + | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/ + ] +] +qed-. + +(* Basic forward lemmas *****************************************************) + +(* Basic_2A1: includes: lift_inv_O2 *) +lemma lifts_fwd_isid: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → 𝐈⦃t⦄ → T1 = T2. +#T1 #T2 #t #H elim H -T1 -T2 -t /4 width=3 by isid_inv_at, eq_f2, eq_f/ +qed-. + +(* Basic_2A1: includes: lift_fwd_pair1 *) +lemma lifts_fwd_pair1: ∀I,V1,T1,Y,t. ⬆*[t] ②{I}V1.T1 ≡ Y → + ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & Y = ②{I}V2.T2. +* [ #a ] #I #V1 #T1 #Y #t #H +[ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/ +| elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/ +] +qed-. + +(* Basic_2A1: includes: lift_fwd_pair2 *) +lemma lifts_fwd_pair2: ∀I,V2,T2,X,t. ⬆*[t] X ≡ ②{I}V2.T2 → + ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & X = ②{I}V1.T1. +* [ #a ] #I #V2 #T2 #X #t #H +[ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/ +| elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/ +] +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: includes: lift_free (right to left) *) +(* Basic_2A1: includes: lift_split *) +lemma lifts_split_trans: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → + ∀t1,t2. t2 ⊚ t1 ≡ t → + ∃∃T. ⬆*[t1] T1 ≡ T & ⬆*[t2] T ≡ T2. +#T1 #T2 #t #H elim H -T1 -T2 -t +[ /3 width=3 by lifts_sort, ex2_intro/ +| #i1 #i2 #t #Hi #t1 #t2 #Ht elim (after_at_fwd … Ht … Hi) -Ht -Hi + /3 width=3 by lifts_lref, ex2_intro/ +| /3 width=3 by lifts_gref, ex2_intro/ +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV #IHT #t1 #t2 #Ht + elim (IHV … Ht) elim (IHT (Ⓣ@t1) (Ⓣ@t2)) -IHV -IHT + /3 width=5 by lifts_bind, after_true, ex2_intro/ +| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV #IHT #t1 #t2 #Ht + elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht + /3 width=5 by lifts_flat, ex2_intro/ +] +qed-. + +(* Note: apparently, this was missing in Basic_2A1 *) +lemma lifts_split_div: ∀T1,T2,t1. ⬆*[t1] T1 ≡ T2 → + ∀t2,t. t2 ⊚ t1 ≡ t → + ∃∃T. ⬆*[t2] T2 ≡ T & ⬆*[t] T1 ≡ T. +#T1 #T2 #t1 #H elim H -T1 -T2 -t1 +[ /3 width=3 by lifts_sort, ex2_intro/ +| #i1 #i2 #t1 #Hi #t2 #t #Ht elim (after_at1_fwd … Ht … Hi) -Ht -Hi + /3 width=3 by lifts_lref, ex2_intro/ +| /3 width=3 by lifts_gref, ex2_intro/ +| #a #I #V1 #V2 #T1 #T2 #t1 #_ #_ #IHV #IHT #t2 #t #Ht + elim (IHV … Ht) elim (IHT (Ⓣ@t2) (Ⓣ@t)) -IHV -IHT + /3 width=5 by lifts_bind, after_true, ex2_intro/ +| #I #V1 #V2 #T1 #T2 #t1 #_ #_ #IHV #IHT #t2 #t #Ht + elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht + /3 width=5 by lifts_flat, ex2_intro/ +] +qed-. + +(* Basic_1: includes: dnf_dec2 dnf_dec *) +(* Basic_2A1: includes: is_lift_dec *) +lemma is_lifts_dec: ∀T2,t. Decidable (∃T1. ⬆*[t] T1 ≡ T2). +#T1 elim T1 -T1 +[ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ] + #i2 #t elim (is_at_dec t i2) + [ * /4 width=3 by lifts_lref, ex_intro, or_introl/ + | #H @or_intror * + #X #HX elim (lifts_inv_lref2 … HX) -HX + /3 width=2 by ex_intro/ + ] +| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #t + [ elim (IHV2 t) -IHV2 + [ * #V1 #HV12 elim (IHT2 (Ⓣ@t)) -IHT2 + [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/ + | -V1 #HT2 @or_intror * #X #H + elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/ + ] + | -IHT2 #HV2 @or_intror * #X #H + elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/ + ] + | elim (IHV2 t) -IHV2 + [ * #V1 #HV12 elim (IHT2 t) -IHT2 + [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/ + | -V1 #HT2 @or_intror * #X #H + elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/ + ] + | -IHT2 #HV2 @or_intror * #X #H + elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/ + ] + ] +] +qed-. + +(* Basic_2A1: removed theorems 17: + lifts_inv_nil lifts_inv_cons lifts_total + lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1 + lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred + lift_lref_ge_minus lift_lref_ge_minus_eq lift_total lift_refl +*) +(* Basic_1: removed theorems 8: + lift_lref_gt + lift_head lift_gen_head + lift_weight_map lift_weight lift_weight_add lift_weight_add_O + lift_tlt_dx +*)