X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fmultiple%2Fcpys.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fmultiple%2Fcpys.ma;h=6f3f953f96ca391bddc4114a9eb9a57bcddd67bc;hb=d2545ffd201b1aa49887313791386add78fa8603;hp=0000000000000000000000000000000000000000;hpb=57ae1762497a5f3ea75740e2908e04adb8642cc2;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma new file mode 100644 index 000000000..6f3f953f9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma @@ -0,0 +1,166 @@ +(**************************************************************************) +(* ___ *) +(* ||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/psubststar_6.ma". +include "basic_2A/substitution/cpy.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +definition cpys: ynat → ynat → relation4 genv lenv term term ≝ + λl,m,G. LTC … (cpy l m G). + +interpretation "context-sensitive extended multiple substritution (term)" + 'PSubstStar G L T1 l m T2 = (cpys l m G L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma cpys_ind: ∀G,L,T1,l,m. ∀R:predicate term. R T1 → + (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → R T → R T2) → + ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T2. +#G #L #T1 #l #m #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. + +lemma cpys_ind_dx: ∀G,L,T2,l,m. ∀R:predicate term. R T2 → + (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → R T → R T1) → + ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T1. +#G #L #T2 #l #m #R #HT2 #IHT2 #T1 #HT12 +@(TC_star_ind_dx … HT2 IHT2 … HT12) // +qed-. + +(* Basic properties *********************************************************) + +lemma cpy_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. +/2 width=1 by inj/ qed. + +lemma cpys_strap1: ∀G,L,T1,T,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. +normalize /2 width=3 by step/ qed-. + +lemma cpys_strap2: ∀G,L,T1,T,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. +normalize /2 width=3 by TC_strap/ qed-. + +lemma lsuby_cpys_trans: ∀G,l,m. lsub_trans … (cpys l m G) (lsuby l m). +/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/ +qed-. + +lemma cpys_refl: ∀G,L,l,m. reflexive … (cpys l m G L). +/2 width=1 by cpy_cpys/ qed. + +lemma cpys_bind: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → + ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → + ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] ⓑ{a,I}V2.T2. +#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2 +[ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/ +| /3 width=5 by cpys_strap1, cpy_bind/ +] +qed. + +lemma cpys_flat: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → + ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → + ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] ⓕ{I}V2.T2. +#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2 +[ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/ +| /3 width=5 by cpys_strap1, cpy_flat/ +qed. + +lemma cpys_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T2 → + ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 → + ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T2. +#G #L #T1 #T2 #l1 #m1 #H #l1 #l2 #Hl21 #Hlm12 @(cpys_ind … H) -T2 +/3 width=7 by cpys_strap1, cpy_weak/ +qed-. + +lemma cpys_weak_top: ∀G,L,T1,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, |L| - l] T2. +#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2 +/3 width=4 by cpys_strap1, cpy_weak_top/ +qed-. + +lemma cpys_weak_full: ∀G,L,T1,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2. +#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2 +/3 width=5 by cpys_strap1, cpy_weak_full/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cpys_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀T1,l,m. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → l + m ≤ lt + mt → + ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #T1 #l #m #HTU1 #Hllt #Hlmlmt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU + elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ♯{T1} ≤ ♯{T2}. +#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2 +/2 width=3 by transitive_le/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +(* Note: this can be derived from cpys_inv_atom1 *) +lemma cpys_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶*[l, m] T2 → T2 = ⋆k. +#G #L #T2 #k #l #m #H @(cpys_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 destruct +>(cpy_inv_sort1 … HT2) -HT2 // +qed-. + +(* Note: this can be derived from cpys_inv_atom1 *) +lemma cpys_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶*[l, m] T2 → T2 = §p. +#G #L #T2 #p #l #m #H @(cpys_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 destruct +>(cpy_inv_gref1 … HT2) -HT2 // +qed-. + +lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & + ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 & + U2 = ⓑ{a,I}V2.T2. +#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2 +[ /2 width=5 by ex3_2_intro/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2 + /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/ +] +qed-. + +lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 & + U2 = ⓕ{I}V2.T2. +#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2 +[ /2 width=5 by ex3_2_intro/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (cpy_inv_flat1 … HU2) -HU2 + /3 width=5 by cpys_strap1, ex3_2_intro/ +] +qed-. + +lemma cpys_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 0] T2 → T1 = T2. +#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 // +qed-. + +lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀l,m:nat. + ⦃G, L⦄ ⊢ U1 ▶*[l, m] U2 → ∀T1. ⬆[l, m] T1 ≡ U1 → U1 = U2. +#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind … H) -U2 +/2 width=7 by cpy_inv_lift1_eq/ +qed-.