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=0000000000000000000000000000000000000000;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=6f3f953f96ca391bddc4114a9eb9a57bcddd67bc;hpb=277fc8ff21ce3dbd6893b1994c55cf5c06a98355;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 deleted file mode 100644 index 6f3f953f9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma +++ /dev/null @@ -1,166 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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-.