From: Ferruccio Guidi Date: Wed, 21 Oct 2015 16:59:38 +0000 (+0000) Subject: theory of multiple relocation completed X-Git-Tag: make_still_working~681 X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=commitdiff_plain;h=7593c0f74b944fb100493fb24b665ce3b8d1d252 theory of multiple relocation completed --- diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys.etc new file mode 100644 index 000000000..7da756f97 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys.etc @@ -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_2/notation/relations/psubststar_6.ma". +include "basic_2/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. + ⦃G, L⦄ ⊢ U1 ▶*[l, yinj 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_alt.etc new file mode 100644 index 000000000..d6b6c3096 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_alt.etc @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/psubststaralt_6.ma". +include "basic_2/multiple/cpys_lift.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +(* alternative definition of cpys *) +inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝ +| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I}) +| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m → + ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(l+m-i)) G K V1 V2 → + ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2 +| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. + cpysa l m G L V1 V2 → cpysa (⫯l) m G (L.ⓑ{I}V1) T1 T2 → + cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) +| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m. + cpysa l m G L V1 V2 → cpysa l m G L T1 T2 → + cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) +. + +interpretation + "context-sensitive extended multiple substritution (term) alternative" + 'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2). + +(* Basic properties *********************************************************) + +lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m). +#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m +[ // +| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 + elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/ +| /4 width=1 by lsuby_succ, cpysa_bind/ +| /3 width=1 by cpysa_flat/ +] +qed-. + +lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T. +#G #T elim T -T // +#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/ +qed. + +lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T → + ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. +#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m +[ #I #G #L #l #m #X #H + elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/ +| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H + lapply (drop_fwd_drop2 … HLK) #H0LK + lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H + elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2 + /3 width=7 by cpysa_subst, ylt_fwd_le_succ1/ +| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H + elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/ +| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H + elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/ +] +qed-. + +lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. +/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. +#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m +/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/ +qed-. + +(* Advanced eliminators *****************************************************) + +lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term. + (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) → + (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m → + ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 → + ⬆[O, i+1] V2 ≡ W2 → R O (⫰(l+m-i)) G K V1 V2 → R l m G L (#i) W2 + ) → + (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → + ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → R l m G L V1 V2 → + R (⫯l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) + ) → + (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 → + R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) + ) → + ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2. +#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m +/3 width=8 by cpysa_inv_cpys/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_cpys.etc new file mode 100644 index 000000000..7df95948e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_cpys.etc @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/substitution/cpy_cpy.ma". +include "basic_2/multiple/cpys_alt.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +(* Advanced inversion lemmas ************************************************) + +lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2. +#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/ +qed-. + +(* Advanced properties ******************************************************) + +lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-. + +lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 → + (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → + ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-. + +lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed. + +lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-. + +lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → + ∀i. l ≤ i → i ≤ l + m → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2. +#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2 +[ /2 width=3 by ex2_intro/ +| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 + elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02 + elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/ + >ymax_pre_sn_comm // +] +qed-. + +lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & + ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt +elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 +lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 +lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct +elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 // +>yplus_minus_inj /2 width=3 by ex2_intro/ +qed-. + +(* Main properties **********************************************************) + +theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-. + +theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 → + (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → + ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-. + +theorem cpys_trans_eq: ∀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 trans_TC/ qed-. + +theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-. + +theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 → + ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2. +#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 // +[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2 + elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ] + [ lapply (cpys_weak_full … HW2) -HW2 #HW2 + lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 // + [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi + #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 // + | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2 + elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2 + /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ -Hli -Hilm + #X #_ #H elim (lift_inv_lref2_be … H) -H /2 width=1 by ylt_inj/ + ] +| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H + #V #T #HV2 #HT2 #H destruct + lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/ +| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H + #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_lift.etc new file mode 100644 index 000000000..59cddfbc9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_lift.etc @@ -0,0 +1,226 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/substitution/cpy_lift.ma". +include "basic_2/multiple/cpys.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +(* Advanced properties ******************************************************) + +lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m. + l ≤ yinj i → i < l + m → + ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(l+m-i)] U1 → + ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2. +#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1 +[ /3 width=5 by cpy_cpys, cpy_subst/ +| #U #U1 #_ #HU1 #IHU #U2 #HU12 + elim (lift_total U 0 (i+1)) #U0 #HU0 + lapply (IHU … HU0) -IHU #H + lapply (drop_fwd_drop2 … HLK) -HLK #HLK + lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 + lapply (cpy_weak … HU02 l m ? ?) -HU02 + [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] + >yplus_O1 ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ1/ +] +qed. + +lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l. + l ≤ yinj i → + ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 → + ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2. +#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12 +@(cpys_subst … HLK … HU12) >yminus_Y_inj // +qed. + +(* Advanced inversion lemmas *************************************************) + +lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 → + T2 = ⓪{I} ∨ + ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m & + ⬇[i] L ≡ K.ⓑ{J}V1 & + ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & + ⬆[O, i+1] V2 ≡ T2 & + I = LRef i. +#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2 +[ /2 width=1 by or_introl/ +| #T #T2 #_ #HT2 * + [ #H destruct + elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ] + | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI + lapply (drop_fwd_drop2 … HLK) #H + elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT + [2,3,4: /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ ] + /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/ + ] +] +qed-. + +lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → + T2 = #i ∨ + ∃∃I,K,V1,V2. l ≤ i & i < l + m & + ⬇[i] L ≡ K.ⓑ{I}V1 & + ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & + ⬆[O, i+1] V2 ≡ T2. +#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ +* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/ +qed-. + +lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 → + T2 = #i ∨ + ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 & + ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2. +#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/ +* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/ +qed-. + +lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → + ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 → + ∀V2. ⬆[O, i+1] V2 ≡ T2 → + ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 + & l ≤ i + & i < l + m. +#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H +[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK /2 width=1 by ylt_inj/ +| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2 + lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct + lapply (drop_mono … HLY … HLK) -L #H destruct + /2 width=1 by and3_intro/ +] +qed-. + +(* Properties on relocation *************************************************) + +lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → + ∀L,U1,s,l,m. lt + mt ≤ l → ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → + ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → + ∀L,U1,s,l,m. lt ≤ l → l ≤ lt + mt → + ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 → + ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → + ∀L,U1,s,l,m. l ≤ lt → ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → + ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +(* Inversion lemmas for relocation ******************************************) + +lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt + mt ≤ l → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l + m ≤ lt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +(* Advanced inversion lemmas on relocation **********************************) + +lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & + ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → lt + mt ≤ l + m → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 → + ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 → + l ≤ yinj i → i < l + m → + ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2. +#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm +elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) // +>yplus_O1 yplus_SO2 +[ >yminus_succ2 /2 width=3 by ex2_intro/ +| /2 width=1 by ylt_fwd_le_succ1/ +| /2 width=3 by yle_trans/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor.etc new file mode 100644 index 000000000..a2d1e61a7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor.etc @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/lazyor_5.ma". +include "basic_2/multiple/frees.ma". + +(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************) + +definition llor: ynat → relation4 term lenv lenv lenv ≝ λl,T,L2,L1,L. + ∧∧ |L1| = |L2| & |L1| = |L| + & (∀I1,I2,I,K1,K2,K,V1,V2,V,i. + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → ⬇[i] L ≡ K.ⓑ{I}V → ∨∨ + (∧∧ yinj i < l & I1 = I & V1 = V) | + (∧∧ (L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → ⊥) & I1 = I & V1 = V) | + (∧∧ l ≤ yinj i & L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ & I2 = I & V2 = V) + ). + +interpretation + "lazy union (local environment)" + 'LazyOr L1 T l L2 L = (llor l T L2 L1 L). + +(* Basic properties *********************************************************) + +(* Note: this can be proved by llor_skip *) +lemma llor_atom: ∀T,l. ⋆ ⋓[T, l] ⋆ ≡ ⋆. +#T #l @and3_intro // +#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1 +elim (drop_inv_atom1 … HLK1) -HLK1 #H destruct +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor_alt.etc new file mode 100644 index 000000000..700cd7ce3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor_alt.etc @@ -0,0 +1,74 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/frees_append.ma". +include "basic_2/multiple/llor.ma". + +(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************) + +(* Alternative definition ***************************************************) + +lemma llor_tail_frees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → l ≤ yinj (|L1|) → + ∀I1,W1. ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → + ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I2}W2.L. +#L1 #L2 #L #U #l * #HL12 #HL1 #IH #Hl #I1 #W1 #HU #I2 #W2 +@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ] +#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK +lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H +lapply (lt_plus_SO_to_le … H) -H #H +elim (le_to_or_lt_eq … H) -H #H +[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1 + elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2 + elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY + lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct + lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct + lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct + elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY * + [ /3 width=1 by and3_intro, or3_intro0/ + | /7 width=2 by frees_inv_append, yle_inj, lt_to_le, or3_intro1, and3_intro/ + | /6 width=1 by frees_append, yle_inj, lt_to_le, or3_intro2, and4_intro/ + ] +| -IH -HLK1 destruct + lapply (drop_O1_inv_append1_le … HLK2 … (⋆) ?) // -HLK2 normalize #H destruct + lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct + /3 width=1 by or3_intro2, and4_intro/ +] +qed. + +lemma llor_tail_cofrees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → + ∀I1,W1. (ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → ⊥) → + ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I1}W1.L. +#L1 #L2 #L #U #l * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2 +@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ] +#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK +lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H +lapply (lt_plus_SO_to_le … H) -H #H +elim (le_to_or_lt_eq … H) -H #H +[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1 + elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2 + elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY + lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct + lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct + lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct + elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY * + [ /3 width=1 by and3_intro, or3_intro0/ + | /7 width=2 by frees_inv_append, yle_inj, lt_to_le, or3_intro1, and3_intro/ + | /6 width=1 by frees_append, yle_inj, lt_to_le, or3_intro2, and4_intro/ + ] +| -IH -HLK2 destruct + lapply (drop_O1_inv_append1_le … HLK1 … (⋆) ?) // -HLK1 normalize #H destruct + lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct + /4 width=1 by or3_intro1, and3_intro/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor_drop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor_drop.etc new file mode 100644 index 000000000..cbeab4a59 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llor/llor_drop.etc @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/frees_lift.ma". +include "basic_2/multiple/llor_alt.ma". + +(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************) + +(* Advanced properties ******************************************************) + +lemma llor_skip: ∀L1,L2,U,l. |L1| = |L2| → yinj (|L1|) ≤ l → L1 ⋓[U, l] L2 ≡ L1. +#L1 #L2 #U #l #HL12 #Hl @and3_intro // -HL12 +#I1 #I2 #I #K1 #K2 #K #W1 #W2 #W #i #HLK1 #_ #HLK -L2 -K2 +lapply (drop_mono … HLK … HLK1) -HLK #H destruct +lapply (drop_fwd_length_lt2 … HLK1) -K1 +/5 width=3 by ylt_yle_trans, ylt_inj, or3_intro0, and3_intro/ +qed. + +(* Note: lemma 1400 concludes the "big tree" theorem *) +lemma llor_total: ∀L1,L2,T,l. |L1| = |L2| → ∃L. L1 ⋓[T, l] L2 ≡ L. +#L1 @(lenv_ind_alt … L1) -L1 +[ #L2 #T #l #H >(length_inv_zero_sn … H) -L2 /2 width=2 by ex_intro/ +| #I1 #L1 #V1 #IHL1 #Y #T #l >ltail_length #H + elim (length_inv_pos_sn_ltail … H) -H #I2 #L2 #V2 #HL12 #H destruct + elim (ylt_split l (|ⓑ{I1}V1.L1|)) + [ elim (frees_dec (ⓑ{I1}V1.L1) T l (|L1|)) #HnU + elim (IHL1 L2 T l) // -IHL1 -HL12 + [ #L #HL12 >ltail_length /4 width=2 by llor_tail_frees, ylt_fwd_succ2, ex_intro/ + | /4 width=2 by llor_tail_cofrees, ex_intro/ + ] + | -IHL1 /4 width=3 by llor_skip, plus_to_minus, ex_intro/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys.ma deleted file mode 100644 index 7da756f97..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/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_2/notation/relations/psubststar_6.ma". -include "basic_2/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. - ⦃G, L⦄ ⊢ U1 ▶*[l, yinj 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-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_alt.ma deleted file mode 100644 index d6b6c3096..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_alt.ma +++ /dev/null @@ -1,102 +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_2/notation/relations/psubststaralt_6.ma". -include "basic_2/multiple/cpys_lift.ma". - -(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) - -(* alternative definition of cpys *) -inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝ -| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I}) -| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m → - ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(l+m-i)) G K V1 V2 → - ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2 -| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. - cpysa l m G L V1 V2 → cpysa (⫯l) m G (L.ⓑ{I}V1) T1 T2 → - cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) -| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m. - cpysa l m G L V1 V2 → cpysa l m G L T1 T2 → - cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) -. - -interpretation - "context-sensitive extended multiple substritution (term) alternative" - 'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2). - -(* Basic properties *********************************************************) - -lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m). -#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m -[ // -| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 - elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/ -| /4 width=1 by lsuby_succ, cpysa_bind/ -| /3 width=1 by cpysa_flat/ -] -qed-. - -lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T. -#G #T elim T -T // -#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/ -qed. - -lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T → - ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. -#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m -[ #I #G #L #l #m #X #H - elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/ -| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H - lapply (drop_fwd_drop2 … HLK) #H0LK - lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H - elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2 - /3 width=7 by cpysa_subst, ylt_fwd_le_succ1/ -| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H - elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct - /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/ -| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H - elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/ -] -qed-. - -lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. -/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/ -qed-. - -(* Advanced eliminators *****************************************************) - -lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term. - (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) → - (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m → - ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 → - ⬆[O, i+1] V2 ≡ W2 → R O (⫰(l+m-i)) G K V1 V2 → R l m G L (#i) W2 - ) → - (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → - ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → R l m G L V1 V2 → - R (⫯l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) - ) → - (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → - ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 → - R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) - ) → - ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2. -#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m -/3 width=8 by cpysa_inv_cpys/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma deleted file mode 100644 index 7df95948e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma +++ /dev/null @@ -1,117 +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_2/substitution/cpy_cpy.ma". -include "basic_2/multiple/cpys_alt.ma". - -(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) - -(* Advanced inversion lemmas ************************************************) - -lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2. -#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/ -qed-. - -(* Advanced properties ******************************************************) - -lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-. - -lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 → - (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → - ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-. - -lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed. - -lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-. - -lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → - ∀i. l ≤ i → i ≤ l + m → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2. -#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2 -[ /2 width=3 by ex2_intro/ -| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 - elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02 - elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/ - >ymax_pre_sn_comm // -] -qed-. - -lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & - ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt -elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 -lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 -lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 // ->yplus_minus_inj /2 width=3 by ex2_intro/ -qed-. - -(* Main properties **********************************************************) - -theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-. - -theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 → - (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → - ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-. - -theorem cpys_trans_eq: ∀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 trans_TC/ qed-. - -theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-. - -theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 → - ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2. -#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 // -[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2 - elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ] - [ lapply (cpys_weak_full … HW2) -HW2 #HW2 - lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 // - [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi - #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 // - | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2 - elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2 - /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ -Hli -Hilm - #X #_ #H elim (lift_inv_lref2_be … H) -H /2 width=1 by ylt_inj/ - ] -| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H - #V #T #HV2 #HT2 #H destruct - lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/ -| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H - #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_lift.ma deleted file mode 100644 index 59cddfbc9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_lift.ma +++ /dev/null @@ -1,226 +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_2/substitution/cpy_lift.ma". -include "basic_2/multiple/cpys.ma". - -(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) - -(* Advanced properties ******************************************************) - -lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m. - l ≤ yinj i → i < l + m → - ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(l+m-i)] U1 → - ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2. -#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1 -[ /3 width=5 by cpy_cpys, cpy_subst/ -| #U #U1 #_ #HU1 #IHU #U2 #HU12 - elim (lift_total U 0 (i+1)) #U0 #HU0 - lapply (IHU … HU0) -IHU #H - lapply (drop_fwd_drop2 … HLK) -HLK #HLK - lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 - lapply (cpy_weak … HU02 l m ? ?) -HU02 - [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] - >yplus_O1 ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ1/ -] -qed. - -lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l. - l ≤ yinj i → - ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 → - ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2. -#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12 -@(cpys_subst … HLK … HU12) >yminus_Y_inj // -qed. - -(* Advanced inversion lemmas *************************************************) - -lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 → - T2 = ⓪{I} ∨ - ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m & - ⬇[i] L ≡ K.ⓑ{J}V1 & - ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & - ⬆[O, i+1] V2 ≡ T2 & - I = LRef i. -#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2 -[ /2 width=1 by or_introl/ -| #T #T2 #_ #HT2 * - [ #H destruct - elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ] - | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI - lapply (drop_fwd_drop2 … HLK) #H - elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT - [2,3,4: /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ ] - /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/ - ] -] -qed-. - -lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → - T2 = #i ∨ - ∃∃I,K,V1,V2. l ≤ i & i < l + m & - ⬇[i] L ≡ K.ⓑ{I}V1 & - ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & - ⬆[O, i+1] V2 ≡ T2. -#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ -* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/ -qed-. - -lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 → - T2 = #i ∨ - ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 & - ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2. -#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/ -* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/ -qed-. - -lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → - ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 → - ∀V2. ⬆[O, i+1] V2 ≡ T2 → - ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 - & l ≤ i - & i < l + m. -#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H -[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK /2 width=1 by ylt_inj/ -| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2 - lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct - lapply (drop_mono … HLY … HLK) -L #H destruct - /2 width=1 by and3_intro/ -] -qed-. - -(* Properties on relocation *************************************************) - -lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → - ∀L,U1,s,l,m. lt + mt ≤ l → ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → - ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → - ∀L,U1,s,l,m. lt ≤ l → l ≤ lt + mt → - ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 → - ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → - ∀L,U1,s,l,m. l ≤ lt → ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → - ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -(* Inversion lemmas for relocation ******************************************) - -lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt + mt ≤ l → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l + m ≤ lt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -(* Advanced inversion lemmas on relocation **********************************) - -lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & - ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → lt + mt ≤ l + m → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 → - ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 → - l ≤ yinj i → i < l + m → - ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2. -#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm -elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) // ->yplus_O1 yplus_SO2 -[ >yminus_succ2 /2 width=3 by ex2_intro/ -| /2 width=1 by ylt_fwd_le_succ1/ -| /2 width=3 by yle_trans/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma deleted file mode 100644 index 065135c9a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma +++ /dev/null @@ -1,124 +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_2/notation/relations/rdropstar_3.ma". -include "basic_2/notation/relations/rdropstar_4.ma". -include "basic_2/substitution/drop.ma". -include "basic_2/multiple/mr2_minus.ma". -include "basic_2/multiple/lifts_vector.ma". - -(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) - -inductive drops (s:bool): list2 ynat 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 ylt_to_minus/ #K #V pluss_SO2 >pluss_SO2 - >yminus_succ2 >ylt_inv_O1 /2 width=1 by ylt_to_minus/ commutative_plus (**) (* (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 *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma deleted file mode 100644 index 2bf29c636..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma +++ /dev/null @@ -1,35 +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_2/substitution/drop_drop.ma". -include "basic_2/multiple/drops.ma". - -(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) - -(* Properties concerning basic local environment slicing ********************) - -lemma drops_drop_trans: ∀L1,L,cs. ⬇*[Ⓕ, cs] L1 ≡ L → ∀L2,i. ⬇[i] L ≡ L2 → - ∃∃L0,cs0,i0. ⬇[i0] L1 ≡ L0 & ⬇*[Ⓕ, cs0] L0 ≡ L2 & - @⦃i, cs⦄ ≡ i0 & cs ▭ i ≡ cs0. -#L1 #L #cs #H elim H -L1 -L -cs -[ /2 width=7 by drops_nil, minuss_nil, at_nil, ex4_3_intro/ -| #L1 #L0 #L #cs #l #m #_ #HL0 #IHL0 #L2 #i #HL2 - elim (ylt_split i l) #Hil - [ elim (drop_trans_le … HL0 … HL2) -L /2 width=2 by ylt_fwd_le/ - #L #HL0 #HL2 elim (IHL0 … HL0) -L0 /3 width=7 by drops_cons, minuss_lt, at_lt, ex4_3_intro/ - | lapply (drop_trans_ge … HL0 … HL2 ?) -L // #HL02 - elim (IHL0 … HL02) -L0 /3 width=7 by minuss_ge, at_ge, ex4_3_intro/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drops.ma deleted file mode 100644 index c2b0b30d1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drops.ma +++ /dev/null @@ -1,25 +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_2/multiple/drops_drop.ma". - -(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: drop1_trans *) -theorem drops_trans: ∀L,L2,s,cs2. ⬇*[s, cs2] L ≡ L2 → ∀L1,cs1. ⬇*[s, cs1] L1 ≡ L → - ⬇*[s, cs2 @@ cs1] L1 ≡ L2. -#L #L2 #s #cs2 #H elim H -L -L2 -cs2 /3 width=3 by drops_cons/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma deleted file mode 100644 index 1ca5528a6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma +++ /dev/null @@ -1,43 +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_2/notation/relations/lazyeq_7.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR CLOSURES ********************************************) - -inductive fleq (l) (G) (L1) (T): relation3 genv lenv term ≝ -| fleq_intro: ∀L2. L1 ≡[T, l] L2 → fleq l G L1 T G L2 T -. - -interpretation - "lazy equivalence (closure)" - 'LazyEq l G1 L1 T1 G2 L2 T2 = (fleq l G1 L1 T1 G2 L2 T2). - -(* Basic properties *********************************************************) - -lemma fleq_refl: ∀l. tri_reflexive … (fleq l). -/2 width=1 by fleq_intro/ qed. - -lemma fleq_sym: ∀l. tri_symmetric … (fleq l). -#l #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma fleq_inv_gen: ∀G1,G2,L1,L2,T1,T2,l. ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄ → - ∧∧ G1 = G2 & L1 ≡[T1, l] L2 & T1 = T2. -#G1 #G2 #L1 #L2 #T1 #T2 #l * -G2 -L2 -T2 /2 width=1 by and3_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma deleted file mode 100644 index 915be044e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma +++ /dev/null @@ -1,34 +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_2/multiple/lleq_lleq.ma". -include "basic_2/multiple/fleq.ma". - -(* LAZY EQUIVALENCE FOR CLOSURES *******************************************) - -(* Main properties **********************************************************) - -theorem fleq_trans: ∀l. tri_transitive … (fleq l). -#l #G1 #G #L1 #L #T1 #T * -G -L -T -#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2 -/3 width=3 by lleq_trans, fleq_intro/ -qed-. - -theorem fleq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2,l. - ⦃G, L, T⦄ ≡[l] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[l] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄. -/3 width=5 by fleq_trans, fleq_sym/ qed-. - -theorem fleq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T,l. - ⦃G1, L1, T1⦄ ≡[l] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[l] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄. -/3 width=5 by fleq_trans, fleq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma deleted file mode 100644 index b126081ba..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma +++ /dev/null @@ -1,109 +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_2/notation/relations/suptermplus_6.ma". -include "basic_2/substitution/fqu.ma". - -(* PLUS-ITERATED SUPCLOSURE *************************************************) - -definition fqup: tri_relation genv lenv term ≝ tri_TC … fqu. - -interpretation "plus-iterated structural successor (closure)" - 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup G1 L1 T1 G2 L2 T2). - -(* Basic properties *********************************************************) - -lemma fqu_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -/2 width=1 by tri_inj/ qed. - -lemma fqup_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -/2 width=5 by tri_step/ qed. - -lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -/2 width=5 by tri_TC_strap/ qed. - -lemma fqup_drop: ∀G1,G2,L1,K1,K2,T1,T2,U1,m. ⬇[m] L1 ≡ K1 → ⬆[0, m] T1 ≡ U1 → - ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄. -#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #m #HLK1 #HTU1 #HT12 elim (eq_or_gt … m) #H destruct -[ >(drop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 // -| /3 width=5 by fqup_strap2, fqu_drop_lt/ -] -qed-. - -lemma fqup_lref: ∀I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄. -/3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_drop/ qed. - -lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+ ⦃G, L, V⦄. -/2 width=1 by fqu_pair_sn, fqu_fqup/ qed. - -lemma fqup_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊐+ ⦃G, L.ⓑ{I}V, T⦄. -/2 width=1 by fqu_bind_dx, fqu_fqup/ qed. - -lemma fqup_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+ ⦃G, L, T⦄. -/2 width=1 by fqu_flat_dx, fqu_fqup/ qed. - -lemma fqup_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+ ⦃G, L, V2⦄. -/2 width=5 by fqu_pair_sn, fqup_strap1/ qed. - -lemma fqup_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I1}V1, T⦄. -/2 width=5 by fqu_flat_dx, fqup_strap1/ qed. - -lemma fqup_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I2}V2, T⦄. -/2 width=5 by fqu_bind_dx, fqup_strap1/ qed. - -(* Basic eliminators ********************************************************) - -lemma fqup_ind: ∀G1,L1,T1. ∀R:relation3 …. - (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2. -#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H -@(tri_TC_ind … IH1 IH2 G2 L2 T2 H) -qed-. - -lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. - (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G1 L1 T1) → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G1 L1 T1. -#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H -@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H) -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2. - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}. -#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 -/3 width=3 by fqu_fwd_fw, transitive_lt/ -qed-. - -(* Advanced eliminators *****************************************************) - -lemma fqup_wf_ind: ∀R:relation3 …. ( - ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) → - R G1 L1 T1 - ) → ∀G1,L1,T1. R G1 L1 T1. -#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/ -qed-. - -lemma fqup_wf_ind_eq: ∀R:relation3 …. ( - ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) → - ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2 - ) → ∀G1,L1,T1. R G1 L1 T1. -#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup_fqup.ma deleted file mode 100644 index 48420a7cf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup_fqup.ma +++ /dev/null @@ -1,22 +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_2/multiple/fqup.ma". - -(* PLUS-ITERATED SUPCLOSURE *************************************************) - -(* Main properties **********************************************************) - -theorem fqup_trans: tri_transitive … fqup. -/2 width=5 by tri_TC_transitive/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma deleted file mode 100644 index 92679770e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma +++ /dev/null @@ -1,83 +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_2/notation/relations/suptermstar_6.ma". -include "basic_2/substitution/fquq.ma". -include "basic_2/multiple/fqup.ma". - -(* STAR-ITERATED SUPCLOSURE *************************************************) - -definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq. - -interpretation "star-iterated structural successor (closure)" - 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2). - -(* Basic eliminators ********************************************************) - -lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2. -#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H -@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) // -qed-. - -lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1. -#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H -@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) // -qed-. - -(* Basic properties *********************************************************) - -lemma fqus_refl: tri_reflexive … fqus. -/2 width=1 by tri_inj/ qed. - -lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. -/2 width=1 by tri_inj/ qed. - -lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. -/2 width=5 by tri_step/ qed-. - -lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. -/2 width=5 by tri_TC_strap/ qed-. - -lemma fqus_drop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ → - ∀L1,U1,m. ⬇[m] L1 ≡ K1 → ⬆[0, m] T1 ≡ U1 → - ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄. -#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2 -/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/ -qed-. - -lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. -#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 -/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/ -qed. - -(* Basic forward lemmas *****************************************************) - -lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}. -#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2 -/3 width=3 by fquq_fwd_fw, transitive_le/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → - ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄. -#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/ -#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus_alt.ma deleted file mode 100644 index c9ec457b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus_alt.ma +++ /dev/null @@ -1,61 +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_2/substitution/fquq_alt.ma". -include "basic_2/multiple/fqus.ma". - -(* STAR-ITERATED SUPCLOSURE *************************************************) - -(* Advanced inversion lemmas ************************************************) - -lemma fqus_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2). -#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 // -#G #G2 #L #L2 #T #T2 #_ #H2 * elim (fquq_inv_gen … H2) -H2 -[ /3 width=5 by fqup_strap1, or_introl/ -| * #HG #HL #HT destruct /2 width=1 by or_introl/ -| #H2 * #HG #HL #HT destruct /3 width=1 by fqu_fqup, or_introl/ -| * #H1G #H1L #H1T * #H2G #H2L #H2T destruct /2 width=1 by or_intror/ -] -qed-. - -(* Advanced properties ******************************************************) - -lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H1) -H1 -[ /2 width=5 by fqup_strap1/ -| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/ -] -qed-. - -lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H2) -H2 -[ /2 width=5 by fqup_strap2/ -| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/ -] -qed-. - -lemma fqus_fqup_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2 -/2 width=5 by fqus_strap1_fqu, fqup_strap1/ -qed-. - -lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → - ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. -#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1 -/3 width=5 by fqus_strap2_fqu, fqup_strap2/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus_fqus.ma deleted file mode 100644 index ef9902931..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus_fqus.ma +++ /dev/null @@ -1,22 +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_2/multiple/fqus.ma". - -(* STAR-ITERATED SUPCLOSURE *************************************************) - -(* Main properties **********************************************************) - -theorem fqus_trans: tri_transitive … fqus. -/2 width=5 by tri_TC_transitive/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma deleted file mode 100644 index 7a091abce..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma +++ /dev/null @@ -1,150 +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_2/notation/relations/rliftstar_3.ma". -include "basic_2/substitution/lift.ma". -include "basic_2/multiple/mr2_plus.ma". - -(* GENERIC TERM RELOCATION **************************************************) - -inductive lifts: list2 ynat nat → relation term ≝ -| lifts_nil : ∀T. lifts (◊) T T -| lifts_cons: ∀T1,T,T2,cs,l,m. - ⬆[l,m] T1 ≡ T → lifts cs T T2 → lifts ({l, m} @ cs) T1 T2 -. - -interpretation "generic relocation (term)" - 'RLiftStar cs T1 T2 = (lifts cs T1 T2). - -(* Basic inversion lemmas ***************************************************) - -fact lifts_inv_nil_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → cs = ◊ → T1 = T2. -#T1 #T2 #cs * -T1 -T2 -cs // -#T1 #T #T2 #l #m #cs #_ #_ #H destruct -qed-. - -lemma lifts_inv_nil: ∀T1,T2. ⬆*[◊] T1 ≡ T2 → T1 = T2. -/2 width=3 by lifts_inv_nil_aux/ qed-. - -fact lifts_inv_cons_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → - ∀l,m,tl. cs = {l, m} @ tl → - ∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[tl] T ≡ T2. -#T1 #T2 #cs * -T1 -T2 -cs -[ #T #l #m #tl #H destruct -| #T1 #T #T2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct - /2 width=3 by ex2_intro/ -qed-. - -lemma lifts_inv_cons: ∀T1,T2,l,m,cs. ⬆*[{l, m} @ cs] T1 ≡ T2 → - ∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[cs] T ≡ T2. -/2 width=3 by lifts_inv_cons_aux/ qed-. - -(* Basic_1: was: lift1_sort *) -lemma lifts_inv_sort1: ∀T2,k,cs. ⬆*[cs] ⋆k ≡ T2 → T2 = ⋆k. -#T2 #k #cs elim cs -cs -[ #H <(lifts_inv_nil … H) -H // -| #l #m #cs #IH #H - elim (lifts_inv_cons … H) -H #X #H - >(lift_inv_sort1 … H) -H /2 width=1 by/ -] -qed-. - -(* Basic_1: was: lift1_lref *) -lemma lifts_inv_lref1: ∀T2,cs,i1. ⬆*[cs] #i1 ≡ T2 → - ∃∃i2. @⦃i1, cs⦄ ≡ i2 & T2 = #i2. -#T2 #cs elim cs -cs -[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/ -| #l #m #cs #IH #i1 #H - elim (lifts_inv_cons … H) -H #X #H1 #H2 - elim (lift_inv_lref1 … H1) -H1 * #Hli1 #H destruct - elim (IH … H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/ -] -qed-. - -lemma lifts_inv_gref1: ∀T2,p,cs. ⬆*[cs] §p ≡ T2 → T2 = §p. -#T2 #p #cs elim cs -cs -[ #H <(lifts_inv_nil … H) -H // -| #l #m #cs #IH #H - elim (lifts_inv_cons … H) -H #X #H - >(lift_inv_gref1 … H) -H /2 width=1 by/ -] -qed-. - -(* Basic_1: was: lift1_bind *) -lemma lifts_inv_bind1: ∀a,I,T2,cs,V1,U1. ⬆*[cs] ⓑ{a,I} V1. U1 ≡ T2 → - ∃∃V2,U2. ⬆*[cs] V1 ≡ V2 & ⬆*[cs + 1] U1 ≡ U2 & - T2 = ⓑ{a,I} V2. U2. -#a #I #T2 #cs elim cs -cs -[ #V1 #U1 #H - <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/ -| #l #m #cs #IHcs #V1 #U1 #H - elim (lifts_inv_cons … H) -H #X #H #HT2 - elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct - elim (IHcs … HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct - /3 width=5 by ex3_2_intro, lifts_cons/ -] -qed-. - -(* Basic_1: was: lift1_flat *) -lemma lifts_inv_flat1: ∀I,T2,cs,V1,U1. ⬆*[cs] ⓕ{I} V1. U1 ≡ T2 → - ∃∃V2,U2. ⬆*[cs] V1 ≡ V2 & ⬆*[cs] U1 ≡ U2 & - T2 = ⓕ{I} V2. U2. -#I #T2 #cs elim cs -cs -[ #V1 #U1 #H - <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/ -| #l #m #cs #IHcs #V1 #U1 #H - elim (lifts_inv_cons … H) -H #X #H #HT2 - elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct - elim (IHcs … HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct - /3 width=5 by ex3_2_intro, lifts_cons/ -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lifts_simple_dx: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. -#T1 #T2 #cs #H elim H -T1 -T2 -cs /3 width=5 by lift_simple_dx/ -qed-. - -lemma lifts_simple_sn: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. -#T1 #T2 #cs #H elim H -T1 -T2 -cs /3 width=5 by lift_simple_sn/ -qed-. - -(* Basic properties *********************************************************) - -lemma lifts_bind: ∀a,I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 → - ∀T1. ⬆*[cs + 1] T1 ≡ T2 → - ⬆*[cs] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2. -#a #I #T2 #V1 #V2 #cs #H elim H -V1 -V2 -cs -[ #V #T1 #H >(lifts_inv_nil … H) -H // -| #V1 #V #V2 #cs #l #m #HV1 #_ #IHV #T1 #H - elim (lifts_inv_cons … H) -H /3 width=3 by lift_bind, lifts_cons/ -] -qed. - -lemma lifts_flat: ∀I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 → - ∀T1. ⬆*[cs] T1 ≡ T2 → - ⬆*[cs] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2. -#I #T2 #V1 #V2 #cs #H elim H -V1 -V2 -cs -[ #V #T1 #H >(lifts_inv_nil … H) -H // -| #V1 #V #V2 #cs #l #m #HV1 #_ #IHV #T1 #H - elim (lifts_inv_cons … H) -H /3 width=3 by lift_flat, lifts_cons/ -] -qed. - -lemma lifts_total: ∀cs,T1. ∃T2. ⬆*[cs] T1 ≡ T2. -#cs elim cs -cs /2 width=2 by lifts_nil, ex_intro/ -#l #m #cs #IH #T1 elim (lift_total T1 l m) -#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_intro/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma deleted file mode 100644 index 2c693baaa..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma +++ /dev/null @@ -1,63 +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 "ground_2/ynat/ynat_max.ma". -include "basic_2/substitution/lift_lift.ma". -include "basic_2/multiple/mr2_minus.ma". -include "basic_2/multiple/lifts.ma". - -(* GENERIC TERM RELOCATION **************************************************) - -(* Properties concerning basic term relocation ******************************) - -(* Basic_1: was: lift1_xhg (right to left) *) -lemma lifts_lift_trans_le: ∀T1,T,cs. ⬆*[cs] T1 ≡ T → ∀T2. ⬆[0, 1] T ≡ T2 → - ∃∃T0. ⬆[0, 1] T1 ≡ T0 & ⬆*[cs + 1] T0 ≡ T2. -#T1 #T #cs #H elim H -T1 -T -cs -[ /2 width=3 by lifts_nil, ex2_intro/ -| #T1 #T3 #T #cs #l #m #HT13 #_ #IHT13 #T2 #HT2 - elim (IHT13 … HT2) -T #T #HT3 #HT2 - elim (lift_trans_le … HT13 … HT3) -T3 /3 width=5 by lifts_cons, ex2_intro/ -] -qed-. - -(* Basic_1: was: lift1_free (right to left) *) -lemma lifts_lift_trans: ∀cs,i,i0. @⦃i, cs⦄ ≡ i0 → - ∀cs0. cs + 1 ▭ i + 1 ≡ cs0 + 1 → - ∀T1,T0. ⬆*[cs0] T1 ≡ T0 → - ∀T2. ⬆[O, i0 + 1] T0 ≡ T2 → - ∃∃T. ⬆[0, i + 1] T1 ≡ T & ⬆*[cs] T ≡ T2. -#cs elim cs -cs -[ #i #x #H1 #cs0 #H2 #T1 #T0 #HT10 #T2 - <(at_inv_nil … H1) -x #HT02 - lapply (minuss_inv_nil1 … H2) -H2 #H - >(pluss_inv_nil2 … H) in HT10; -cs0 #H - >(lifts_inv_nil … H) -T1 /2 width=3 by lifts_nil, ex2_intro/ -| #l #m #cs #IHcs #i #i0 #H1 #cs0 #H2 #T1 #T0 #HT10 #T2 #HT02 - elim (at_inv_cons … H1) -H1 * #Hil #Hi0 - [ elim (minuss_inv_cons1_lt … H2) -H2 /2 width=1 by ylt_succ/ #cs1 #Hcs1 - yplus_SO2 >yplus_SO2 >yminus_succ #H - elim (pluss_inv_cons2 … H) -H #cs2 #H1 #H2 destruct - elim (lifts_inv_cons … HT10) -HT10 #T #HT1 #HT0 - elim (IHcs … Hi0 … Hcs1 … HT0 … HT02) -IHcs -Hi0 -Hcs1 -T0 #T0 #HT0 #HT02 - elim (lift_trans_le … HT1 … HT0) -T /2 width=1 by/ #T #HT1 - yplus_SO2 >ymax_pre_sn - /3 width=5 by lifts_cons, ex2_intro, ylt_fwd_le_succ1/ - | >commutative_plus in Hi0; #Hi0 - lapply (minuss_inv_cons1_ge … H2 ?) -H2 /2 width=1 by yle_succ/ (liftv_inv_nil1 … H) -T1s /2 width=3 by liftsv_nil, liftv_nil, ex2_intro/ -| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H - elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct - elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s - elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5 by liftsv_cons, liftv_cons, ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lifts.ma deleted file mode 100644 index cff6bf238..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lifts.ma +++ /dev/null @@ -1,25 +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_2/multiple/lifts_lift.ma". - -(* GENERIC RELOCATION *******************************************************) - -(* Main properties **********************************************************) - -(* Basic_1: was: lift1_lift1 (left to right) *) -theorem lifts_trans: ∀T1,T,cs1. ⬆*[cs1] T1 ≡ T → ∀T2:term. ∀cs2. ⬆*[cs2] T ≡ T2 → - ⬆*[cs1 @@ cs2] T1 ≡ T2. -#T1 #T #cs1 #H elim H -T1 -T -cs1 /3 width=3 by lifts_cons/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_vector.ma deleted file mode 100644 index f98062e1e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_vector.ma +++ /dev/null @@ -1,53 +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_2/substitution/lift_vector.ma". -include "basic_2/multiple/lifts.ma". - -(* GENERIC TERM VECTOR RELOCATION *******************************************) - -inductive liftsv (cs:list2 ynat nat) : relation (list term) ≝ -| liftsv_nil : liftsv cs (◊) (◊) -| liftsv_cons: ∀T1s,T2s,T1,T2. - ⬆*[cs] T1 ≡ T2 → liftsv cs T1s T2s → - liftsv cs (T1 @ T1s) (T2 @ T2s) -. - -interpretation "generic relocation (vector)" - 'RLiftStar cs T1s T2s = (liftsv cs T1s T2s). - -(* Basic inversion lemmas ***************************************************) - -(* Basic_1: was: lifts1_flat (left to right) *) -lemma lifts_inv_applv1: ∀V1s,U1,T2,cs. ⬆*[cs] Ⓐ V1s. U1 ≡ T2 → - ∃∃V2s,U2. ⬆*[cs] V1s ≡ V2s & ⬆*[cs] U1 ≡ U2 & - T2 = Ⓐ V2s. U2. -#V1s elim V1s -V1s normalize -[ #T1 #T2 #cs #HT12 - @ex3_2_intro [3,4: // |1,2: skip | // ] (**) (* explicit constructor *) -| #V1 #V1s #IHV1s #T1 #X #cs #H - elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct - elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct - @(ex3_2_intro) [4: // |3: /2 width=2 by liftsv_cons/ |1,2: skip | // ] (**) (* explicit constructor *) -] -qed-. - -(* Basic properties *********************************************************) - -(* Basic_1: was: lifts1_flat (right to left) *) -lemma lifts_applv: ∀V1s,V2s,cs. ⬆*[cs] V1s ≡ V2s → - ∀T1,T2. ⬆*[cs] T1 ≡ T2 → - ⬆*[cs] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2. -#V1s #V2s #cs #H elim H -V1s -V2s /3 width=1 by lifts_flat/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma deleted file mode 100644 index 1f57f73be..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma +++ /dev/null @@ -1,160 +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_2/notation/relations/lazyeq_4.ma". -include "basic_2/multiple/llpx_sn.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2. - -definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq. - -interpretation - "lazy equivalence (local environment)" - 'LazyEq T l L1 L2 = (lleq l T L1 L2). - -definition lleq_transitive: predicate (relation3 lenv term term) ≝ - λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2. - -(* Basic inversion lemmas ***************************************************) - -lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. ( - ∀L1,L2,l,k. |L1| = |L2| → R l (⋆k) L1 L2 - ) → ( - ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2 - ) → ( - ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → - ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2 - ) → ( - ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2 - ) → ( - ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2 - ) → ( - ∀a,I,L1,L2,V,T,l. - L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → - R l V L1 L2 → R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2 - ) → ( - ∀I,L1,L2,V,T,l. - L1 ≡[V, l]L2 → L1 ≡[T, l] L2 → - R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2 - ) → - ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2. -#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/ -qed-. - -lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,l. L1 ≡[ⓑ{a,I}V.T, l] L2 → - L1 ≡[V, l] L2 ∧ L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V. -/2 width=2 by llpx_sn_inv_bind/ qed-. - -lemma lleq_inv_flat: ∀I,L1,L2,V,T,l. L1 ≡[ⓕ{I}V.T, l] L2 → - L1 ≡[V, l] L2 ∧ L1 ≡[T, l] L2. -/2 width=2 by llpx_sn_inv_flat/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lleq_fwd_length: ∀L1,L2,T,l. L1 ≡[T, l] L2 → |L1| = |L2|. -/2 width=4 by llpx_sn_fwd_length/ qed-. - -lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → - ∨∨ |L1| ≤ i ∧ |L2| ≤ i - | yinj i < l - | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V & - ⬇[i] L2 ≡ K2.ⓑ{I}V & - K1 ≡[V, yinj 0] K2 & l ≤ yinj i. -#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/ -* /3 width=7 by or3_intro2, ex4_4_intro/ -qed-. - -lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 → - ∃K2. ⬇[i] L2 ≡ K2. -/2 width=7 by llpx_sn_fwd_drop_sn/ qed-. - -lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 → - ∃K1. ⬇[i] L1 ≡ K1. -/2 width=7 by llpx_sn_fwd_drop_dx/ qed-. - -lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,l. - L1 ≡[ⓑ{a,I}V.T, l] L2 → L1 ≡[V, l] L2. -/2 width=4 by llpx_sn_fwd_bind_sn/ qed-. - -lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,l. - L1 ≡[ⓑ{a,I}V.T, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V. -/2 width=2 by llpx_sn_fwd_bind_dx/ qed-. - -lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,l. - L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[V, l] L2. -/2 width=3 by llpx_sn_fwd_flat_sn/ qed-. - -lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,l. - L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[T, l] L2. -/2 width=3 by llpx_sn_fwd_flat_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma lleq_sort: ∀L1,L2,l,k. |L1| = |L2| → L1 ≡[⋆k, l] L2. -/2 width=1 by llpx_sn_sort/ qed. - -lemma lleq_skip: ∀L1,L2,l,i. yinj i < l → |L1| = |L2| → L1 ≡[#i, l] L2. -/2 width=1 by llpx_sn_skip/ qed. - -lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → - ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2. -/2 width=9 by llpx_sn_lref/ qed. - -lemma lleq_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, l] L2. -/2 width=1 by llpx_sn_free/ qed. - -lemma lleq_gref: ∀L1,L2,l,p. |L1| = |L2| → L1 ≡[§p, l] L2. -/2 width=1 by llpx_sn_gref/ qed. - -lemma lleq_bind: ∀a,I,L1,L2,V,T,l. - L1 ≡[V, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → - L1 ≡[ⓑ{a,I}V.T, l] L2. -/2 width=1 by llpx_sn_bind/ qed. - -lemma lleq_flat: ∀I,L1,L2,V,T,l. - L1 ≡[V, l] L2 → L1 ≡[T, l] L2 → L1 ≡[ⓕ{I}V.T, l] L2. -/2 width=1 by llpx_sn_flat/ qed. - -lemma lleq_refl: ∀l,T. reflexive … (lleq l T). -/2 width=1 by llpx_sn_refl/ qed. - -lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2. -/2 width=1 by llpx_sn_Y/ qed. - -lemma lleq_sym: ∀l,T. symmetric … (lleq l T). -#l #T #L1 #L2 #H @(lleq_ind … H) -l -T -L1 -L2 -/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/ -qed-. - -lemma lleq_ge_up: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀T,l,m. ⬆[l, m] T ≡ U → - lt ≤ l + m → L1 ≡[U, l] L2. -/2 width=6 by llpx_sn_ge_up/ qed-. - -lemma lleq_ge: ∀L1,L2,T,l1. L1 ≡[T, l1] L2 → ∀l2. l1 ≤ l2 → L1 ≡[T, l2] L2. -/2 width=3 by llpx_sn_ge/ qed-. - -lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → - L1 ≡[ⓑ{a,I}V.T, 0] L2. -/2 width=1 by llpx_sn_bind_O/ qed-. - -(* Advanceded properties on lazy pointwise extensions ************************) - -lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → llpx_sn R l T L1 L2. -/2 width=3 by llpx_sn_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma deleted file mode 100644 index d2919f5b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma +++ /dev/null @@ -1,41 +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_2/multiple/llpx_sn_alt.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Alternative definition (not recursive) ***********************************) - -theorem lleq_intro_alt: ∀L1,L2,T,l. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ V1 = V2 - ) → L1 ≡[T, l] L2. -#L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // -qed. - -theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ V1 = V2. -#L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H -#HL12 #IH @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma deleted file mode 100644 index 440e0f510..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma +++ /dev/null @@ -1,54 +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_2/multiple/llpx_sn_alt_rec.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Alternative definition (recursive) ***************************************) - -theorem lleq_intro_alt_r: ∀L1,L2,T,l. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 - ) → L1 ≡[T, l] L2. -#L1 #L2 #T #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ -qed. - -theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv. - (∀L1,L2,T,l. |L1| = |L2| → ( - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2 - ) → S l T L1 L2) → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → S l T L1 L2. -#S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -l -#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/ -qed-. - -theorem lleq_inv_alt_r: ∀L1,L2,T,l. L1 ≡[T, l] L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2. -#L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r … H) -H -#HL12 #IH @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma deleted file mode 100644 index 09ee8a495..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma +++ /dev/null @@ -1,150 +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_2/multiple/llpx_sn_drop.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Advanced properties ******************************************************) - -lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → - ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W. -/2 width=7 by llpx_sn_bind_repl_O/ qed-. - -lemma lleq_dec: ∀T,L1,L2,l. Decidable (L1 ≡[T, l] L2). -/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-. - -lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → - ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L. -#R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l -[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/ -|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/ -| #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2 - /3 width=11 by llpx_sn_lref/ -| #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H - /3 width=1 by llpx_sn_bind/ -| #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H - /3 width=1 by llpx_sn_flat/ -] -qed-. - -lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → - ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L. -/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → - ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2. -#L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2 -/2 width=3 by ex2_intro/ -qed-. - -lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → - ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2. -#L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1 -/2 width=3 by ex2_intro/ -qed-. - -lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I1,I2,K1,K2,V1,V2. - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2. -/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. - -lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, 0] K2. -#L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2 -elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) // -qed-. - -lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 → - ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, 0] K2 → L1 ≡[T, l] L2. -/2 width=9 by llpx_sn_inv_S/ qed-. - -lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → - L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. -/2 width=2 by llpx_sn_inv_bind_O/ qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → - ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → - i < l ∨ - ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. -#L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 -[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ -qed-. - -lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → - ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → - i < l ∨ - ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. -#L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 -[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ -qed-. - -lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → - L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. -/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-. - -(* Properties on relocation *************************************************) - -lemma lleq_lift_le: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → - ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀U. ⬆[l, m] T ≡ U → lt ≤ l → L1 ≡[U, lt] L2. -/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-. - -lemma lleq_lift_ge: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → - ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀U. ⬆[l, m] T ≡ U → l ≤ lt → L1 ≡[U, lt+m] L2. -/2 width=9 by llpx_sn_lift_ge/ qed-. - -(* Inversion lemmas on relocation *******************************************) - -lemma lleq_inv_lift_le: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀T. ⬆[l, m] T ≡ U → lt ≤ l → K1 ≡[T, lt] K2. -/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-. - -lemma lleq_inv_lift_be: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀T. ⬆[l, m] T ≡ U → l ≤ lt → lt ≤ l + m → K1 ≡[T, l] K2. -/2 width=11 by llpx_sn_inv_lift_be/ qed-. - -lemma lleq_inv_lift_ge: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀T. ⬆[l, m] T ≡ U → l + m ≤ lt → K1 ≡[T, lt-m] K2. -/2 width=9 by llpx_sn_inv_lift_ge/ qed-. - -(* Inversion lemmas on negated lazy quivalence for local environments *******) - -lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) → - (L1 ≡[V, l] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → ⊥). -/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-. - -lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) → - (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥). -/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-. - -lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) → - (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥). -/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma deleted file mode 100644 index 28bb175ed..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma +++ /dev/null @@ -1,75 +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_2/multiple/fqus_alt.ma". -include "basic_2/multiple/lleq_drop.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Properties on supclosure *************************************************) - -lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U -[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H // - #K1 #H1 #H2 lapply (drop_inv_O2 … H1) -H1 - #H destruct /2 width=3 by fqu_lref_O, ex2_intro/ -| * [ #a ] #I #G #L2 #V #T #L1 #H - [ elim (lleq_inv_bind … H) - | elim (lleq_inv_flat … H) - ] -H - /2 width=3 by fqu_pair_sn, ex2_intro/ -| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H - #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/ -| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H - /2 width=3 by fqu_flat_dx, ex2_intro/ -| #G #L2 #K2 #T #U #m #HLK2 #HTU #L1 #HL12 - elim (drop_O1_le (Ⓕ) (m+1) L1) - [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/ - | lapply (drop_fwd_length_le2 … HLK2) -K2 - lapply (lleq_fwd_length … HL12) -T -U // - ] -] -qed-. - -lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H -[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/ -| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ -] -qed-. - -lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U -[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2 - /3 width=3 by fqu_fqup, ex2_intro/ -| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2 - #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K - /3 width=5 by fqup_strap1, ex2_intro/ -] -qed-. - -lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H -[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/ -| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma deleted file mode 100644 index 630b41a32..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma +++ /dev/null @@ -1,39 +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_2/multiple/lleq_drop.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Main properties **********************************************************) - -theorem lleq_trans: ∀l,T. Transitive … (lleq l T). -/2 width=3 by lleq_llpx_sn_trans/ qed-. - -theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2. -/3 width=3 by lleq_trans, lleq_sym/ qed-. - -theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2. -/3 width=3 by lleq_trans, lleq_sym/ qed-. - -(* Advanced properies on negated lazy equivalence *****************************) - -(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *) -lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L → - ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥). -/3 width=3 by lleq_canc_sn/ qed-. - -lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L → - ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥). -/3 width=3 by lleq_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma deleted file mode 100644 index 06893aad6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma +++ /dev/null @@ -1,41 +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_2/multiple/llor.ma". -include "basic_2/multiple/llpx_sn_frees.ma". -include "basic_2/multiple/lleq_alt.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Properties on pointwise union for local environments **********************) - -lemma llpx_sn_llor_dx: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → - ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L2 ≡[T, l] L. -#R #H1R #H2R #L1 #L2 #T #l #H1 #L #H2 -lapply (llpx_sn_frees_trans … H1R H2R … H1) -H1R -H2R #HR -elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1 -elim H2 -H2 #_ #HL1 #IH2 -@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK -lapply (drop_fwd_length_lt2 … HLK) #HiL -elim (drop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1 -elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct -elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H -[ elim (ylt_yle_false … H) -H // -| elim H -H /2 width=1 by/ -] -qed. - -lemma llpx_sn_llor_dx_sym: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → - ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L ≡[T, l] L2. -/3 width=6 by llpx_sn_llor_dx, lleq_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lreq.ma deleted file mode 100644 index 8fa21a5da..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lreq.ma +++ /dev/null @@ -1,36 +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_2/multiple/llpx_sn_lreq.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Properties on equivalence for local environments *************************) - -lemma lreq_lleq_trans: ∀L2,L,T,l. L2 ≡[T, l] L → - ∀L1. L1 ⩬[l, ∞] L2 → L1 ≡[T, l] L. -/2 width=3 by lreq_llpx_sn_trans/ qed-. - -lemma lleq_lreq_trans: ∀L,L1,T,l. L ≡[T, l] L1 → - ∀L2. L1 ⩬[l, ∞] L2 → L ≡[T, l] L2. -/2 width=3 by llpx_sn_lreq_trans/ qed-. - -lemma lleq_lreq_repl: ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀K1. K1 ⩬[l, ∞] L1 → - ∀K2. L2 ⩬[l, ∞] K2 → K1 ≡[T, l] K2. -/2 width=5 by llpx_sn_lreq_repl/ qed-. - -lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 → - ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2. -/2 width=5 by llpx_sn_bind_repl_SO/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma deleted file mode 100644 index a2d1e61a7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma +++ /dev/null @@ -1,40 +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_2/notation/relations/lazyor_5.ma". -include "basic_2/multiple/frees.ma". - -(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************) - -definition llor: ynat → relation4 term lenv lenv lenv ≝ λl,T,L2,L1,L. - ∧∧ |L1| = |L2| & |L1| = |L| - & (∀I1,I2,I,K1,K2,K,V1,V2,V,i. - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → ⬇[i] L ≡ K.ⓑ{I}V → ∨∨ - (∧∧ yinj i < l & I1 = I & V1 = V) | - (∧∧ (L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → ⊥) & I1 = I & V1 = V) | - (∧∧ l ≤ yinj i & L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ & I2 = I & V2 = V) - ). - -interpretation - "lazy union (local environment)" - 'LazyOr L1 T l L2 L = (llor l T L2 L1 L). - -(* Basic properties *********************************************************) - -(* Note: this can be proved by llor_skip *) -lemma llor_atom: ∀T,l. ⋆ ⋓[T, l] ⋆ ≡ ⋆. -#T #l @and3_intro // -#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1 -elim (drop_inv_atom1 … HLK1) -HLK1 #H destruct -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma deleted file mode 100644 index 700cd7ce3..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma +++ /dev/null @@ -1,74 +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_2/multiple/frees_append.ma". -include "basic_2/multiple/llor.ma". - -(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************) - -(* Alternative definition ***************************************************) - -lemma llor_tail_frees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → l ≤ yinj (|L1|) → - ∀I1,W1. ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → - ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I2}W2.L. -#L1 #L2 #L #U #l * #HL12 #HL1 #IH #Hl #I1 #W1 #HU #I2 #W2 -@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ] -#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK -lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H -lapply (lt_plus_SO_to_le … H) -H #H -elim (le_to_or_lt_eq … H) -H #H -[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1 - elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2 - elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY - lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct - lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct - lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct - elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY * - [ /3 width=1 by and3_intro, or3_intro0/ - | /7 width=2 by frees_inv_append, yle_inj, lt_to_le, or3_intro1, and3_intro/ - | /6 width=1 by frees_append, yle_inj, lt_to_le, or3_intro2, and4_intro/ - ] -| -IH -HLK1 destruct - lapply (drop_O1_inv_append1_le … HLK2 … (⋆) ?) // -HLK2 normalize #H destruct - lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct - /3 width=1 by or3_intro2, and4_intro/ -] -qed. - -lemma llor_tail_cofrees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → - ∀I1,W1. (ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → ⊥) → - ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I1}W1.L. -#L1 #L2 #L #U #l * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2 -@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ] -#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK -lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H -lapply (lt_plus_SO_to_le … H) -H #H -elim (le_to_or_lt_eq … H) -H #H -[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1 - elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2 - elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY - lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct - lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct - lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct - elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY * - [ /3 width=1 by and3_intro, or3_intro0/ - | /7 width=2 by frees_inv_append, yle_inj, lt_to_le, or3_intro1, and3_intro/ - | /6 width=1 by frees_append, yle_inj, lt_to_le, or3_intro2, and4_intro/ - ] -| -IH -HLK2 destruct - lapply (drop_O1_inv_append1_le … HLK1 … (⋆) ?) // -HLK1 normalize #H destruct - lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct - /4 width=1 by or3_intro1, and3_intro/ -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma deleted file mode 100644 index cbeab4a59..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma +++ /dev/null @@ -1,45 +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_2/multiple/frees_lift.ma". -include "basic_2/multiple/llor_alt.ma". - -(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************) - -(* Advanced properties ******************************************************) - -lemma llor_skip: ∀L1,L2,U,l. |L1| = |L2| → yinj (|L1|) ≤ l → L1 ⋓[U, l] L2 ≡ L1. -#L1 #L2 #U #l #HL12 #Hl @and3_intro // -HL12 -#I1 #I2 #I #K1 #K2 #K #W1 #W2 #W #i #HLK1 #_ #HLK -L2 -K2 -lapply (drop_mono … HLK … HLK1) -HLK #H destruct -lapply (drop_fwd_length_lt2 … HLK1) -K1 -/5 width=3 by ylt_yle_trans, ylt_inj, or3_intro0, and3_intro/ -qed. - -(* Note: lemma 1400 concludes the "big tree" theorem *) -lemma llor_total: ∀L1,L2,T,l. |L1| = |L2| → ∃L. L1 ⋓[T, l] L2 ≡ L. -#L1 @(lenv_ind_alt … L1) -L1 -[ #L2 #T #l #H >(length_inv_zero_sn … H) -L2 /2 width=2 by ex_intro/ -| #I1 #L1 #V1 #IHL1 #Y #T #l >ltail_length #H - elim (length_inv_pos_sn_ltail … H) -H #I2 #L2 #V2 #HL12 #H destruct - elim (ylt_split l (|ⓑ{I1}V1.L1|)) - [ elim (frees_dec (ⓑ{I1}V1.L1) T l (|L1|)) #HnU - elim (IHL1 L2 T l) // -IHL1 -HL12 - [ #L #HL12 >ltail_length /4 width=2 by llor_tail_frees, ylt_fwd_succ2, ex_intro/ - | /4 width=2 by llor_tail_cofrees, ex_intro/ - ] - | -IHL1 /4 width=3 by llor_skip, plus_to_minus, ex_intro/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma deleted file mode 100644 index cf6e4ad76..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma +++ /dev/null @@ -1,209 +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 "ground_2/ynat/ynat_plus.ma". -include "basic_2/substitution/drop.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝ -| llpx_sn_sort: ∀L1,L2,l,k. |L1| = |L2| → llpx_sn R l (⋆k) L1 L2 -| llpx_sn_skip: ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → llpx_sn R l (#i) L1 L2 -| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i → - ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 → - llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2 -| llpx_sn_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R l (#i) L1 L2 -| llpx_sn_gref: ∀L1,L2,l,p. |L1| = |L2| → llpx_sn R l (§p) L1 L2 -| llpx_sn_bind: ∀a,I,L1,L2,V,T,l. - llpx_sn R l V L1 L2 → llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → - llpx_sn R l (ⓑ{a,I}V.T) L1 L2 -| llpx_sn_flat: ∀I,L1,L2,V,T,l. - llpx_sn R l V L1 L2 → llpx_sn R l T L1 L2 → llpx_sn R l (ⓕ{I}V.T) L1 L2 -. - -(* Basic inversion lemmas ***************************************************) - -fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → - ∀a,I,V,T. X = ⓑ{a,I}V.T → - llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). -#R #L1 #L2 #X #l * -L1 -L2 -X -l -[ #L1 #L2 #l #k #_ #b #J #W #U #H destruct -| #L1 #L2 #l #i #_ #_ #b #J #W #U #H destruct -| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct -| #L1 #L2 #l #i #_ #_ #_ #b #J #W #U #H destruct -| #L1 #L2 #l #p #_ #b #J #W #U #H destruct -| #a #I #L1 #L2 #V #T #l #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/ -| #I #L1 #L2 #V #T #l #_ #_ #b #J #W #U #H destruct -] -qed-. - -lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → - llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). -/2 width=4 by llpx_sn_inv_bind_aux/ qed-. - -fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → - ∀I,V,T. X = ⓕ{I}V.T → - llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2. -#R #L1 #L2 #X #l * -L1 -L2 -X -l -[ #L1 #L2 #l #k #_ #J #W #U #H destruct -| #L1 #L2 #l #i #_ #_ #J #W #U #H destruct -| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #J #W #U #H destruct -| #L1 #L2 #l #i #_ #_ #_ #J #W #U #H destruct -| #L1 #L2 #l #p #_ #J #W #U #H destruct -| #a #I #L1 #L2 #V #T #l #_ #_ #J #W #U #H destruct -| #I #L1 #L2 #V #T #l #HV #HT #J #W #U #H destruct /2 width=1 by conj/ -] -qed-. - -lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 → - llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2. -/2 width=4 by llpx_sn_inv_flat_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma llpx_sn_fwd_length: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → |L1| = |L2|. -#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l // -#I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #HLK1 #HLK2 #_ #_ #HK12 -lapply (drop_fwd_length … HLK1) -HLK1 -lapply (drop_fwd_length … HLK2) -HLK2 -normalize // -qed-. - -lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → - ∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2. -#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H -#HL12 lapply (drop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by drop_O1_le/ -qed-. - -lemma llpx_sn_fwd_drop_dx: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → - ∀K2,i. ⬇[i] L2 ≡ K2 → ∃K1. ⬇[i] L1 ≡ K1. -#R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H -#HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/ -qed-. - -fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → ∀i. X = #i → - ∨∨ |L1| ≤ i ∧ |L2| ≤ i - | yinj i < l - | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & - ⬇[i] L2 ≡ K2.ⓑ{I}V2 & - llpx_sn R (yinj 0) V1 K1 K2 & - R K1 V1 V2 & l ≤ yinj i. -#R #L1 #L2 #X #l * -L1 -L2 -X -l -[ #L1 #L2 #l #k #_ #j #H destruct -| #L1 #L2 #l #i #_ #Hil #j #H destruct /2 width=1 by or3_intro1/ -| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 #j #H destruct - /3 width=9 by or3_intro2, ex5_5_intro/ -| #L1 #L2 #l #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/ -| #L1 #L2 #l #p #_ #j #H destruct -| #a #I #L1 #L2 #V #T #l #_ #_ #j #H destruct -| #I #L1 #L2 #V #T #l #_ #_ #j #H destruct -] -qed-. - -lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → - ∨∨ |L1| ≤ i ∧ |L2| ≤ i - | yinj i < l - | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & - ⬇[i] L2 ≡ K2.ⓑ{I}V2 & - llpx_sn R (yinj 0) V1 K1 K2 & - R K1 V1 V2 & l ≤ yinj i. -/2 width=3 by llpx_sn_fwd_lref_aux/ qed-. - -lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → - llpx_sn R l V L1 L2. -#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H // -qed-. - -lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → - llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). -#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H // -qed-. - -lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 → - llpx_sn R l V L1 L2. -#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H // -qed-. - -lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 → - llpx_sn R l T L1 L2. -#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H // -qed-. - -lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (②{I}V.T) L1 L2 → - llpx_sn R l V L1 L2. -#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/ -qed-. - -(* Basic properties *********************************************************) - -lemma llpx_sn_refl: ∀R. (∀L. reflexive … (R L)) → ∀T,L,l. llpx_sn R l T L L. -#R #HR #T #L @(f2_ind … rfw … L T) -L -T -#x #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/ -#i #Hx elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/ -#HiL #l elim (ylt_split i l) /2 width=1 by llpx_sn_skip/ -elim (drop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, drop_fwd_rfw/ -qed-. - -lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2. -#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T -#x #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/ -#a #I #V1 #T1 #Hx #L2 #HL12 -@llpx_sn_bind /2 width=1 by/ (**) (* explicit constructor *) -@IH -IH // normalize /2 width=1 by eq_f2/ -qed-. - -lemma llpx_sn_ge_up: ∀R,L1,L2,U,lt. llpx_sn R lt U L1 L2 → ∀T,l,m. ⬆[l, m] T ≡ U → - lt ≤ l + m → llpx_sn R l U L1 L2. -#R #L1 #L2 #U #lt #H elim H -L1 -L2 -U -lt -[ #L1 #L2 #lt #k #HL12 #X #l #m #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #lt #i #HL12 #Hilt #X #l #m #H #Hltlm - elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12 - elim (ylt_yle_false … Hilt) -Hilt - @(yle_trans … Hltlm) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *) -| #I #L1 #L2 #K1 #K2 #W1 #W2 #lt #i #Hlti #HLK1 #HLK2 #HW1 #HW12 #_ #X #l #m #H #_ - elim (lift_inv_lref2 … H) -H * #Hil #H destruct - [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12 - lapply (drop_fwd_length … HLK1) lapply (drop_fwd_length … HLK2) - normalize in ⊢ (%→%→?); -I -W1 -W2 -lt /3 width=1 by llpx_sn_skip, ylt_inj/ - | /3 width=9 by llpx_sn_lref, yle_fwd_plus_sn1/ - ] -| /2 width=1 by llpx_sn_free/ -| #L1 #L2 #lt #p #HL12 #X #l #m #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct - elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct - @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *) - @(IHT … HTU) /2 width=1 by yle_succ/ -| #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct - elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct - /3 width=4 by llpx_sn_flat/ -] -qed-. - -(**) (* the minor premise comes first *) -lemma llpx_sn_ge: ∀R,L1,L2,T,l1,l2. l1 ≤ l2 → - llpx_sn R l1 T L1 L2 → llpx_sn R l2 T L1 L2. -#R #L1 #L2 #T #l1 #l2 * -l1 -l2 (**) (* destructed yle *) -/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/ -qed-. - -lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 → - llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → - llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2. -/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-. - -lemma llpx_sn_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → - ∀L1,L2,T,l. llpx_sn R1 l T L1 L2 → llpx_sn R2 l T L1 L2. -#R1 #R2 #HR12 #L1 #L2 #T #l #H elim H -L1 -L2 -T -l -/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma deleted file mode 100644 index f40046c5f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma +++ /dev/null @@ -1,63 +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_2/multiple/frees.ma". -include "basic_2/multiple/llpx_sn_alt_rec.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -(* alternative definition of llpx_sn (not recursive) *) -definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝ - λR,l,T,L1,L2. |L1| = |L2| ∧ - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ R K1 V1 V2 - ). - -(* Main properties **********************************************************) - -theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,l. llpx_sn R l T L1 L2 → llpx_sn_alt R l T L1 L2. -#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U -#x #IHx #L1 #U #Hx #L2 #l #H elim (llpx_sn_inv_alt_r … H) -H -#HL12 #IHU @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 elim (frees_inv … H) -H -[ -x #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/ -| * #J1 #K10 #W10 #j #Hlj #Hji #HnU #HLK10 yminus_inj >yminus_inj #HnW10 destruct - lapply (drop_fwd_drop2 … HLK10) #H - lapply (drop_conf_ge … H … HLK1 ?) -H /2 width=1 by ylt_fwd_le_succ1/ (minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus yminus_SO2 -#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *) -lapply (drop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1 -lapply (drop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2 -/4 width=9 by frees_be, yle_plus_dx2_trans, yle_succ_dx, ylt_inj/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma deleted file mode 100644 index 870a4bbef..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma +++ /dev/null @@ -1,249 +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_2/substitution/lift_neg.ma". -include "basic_2/substitution/drop_drop.ma". -include "basic_2/multiple/llpx_sn.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -(* alternative definition of llpx_sn (recursive) *) -inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝ -| llpx_sn_alt_r_intro: ∀L1,L2,T,l. - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2 - ) → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2 - ) → |L1| = |L2| → llpx_sn_alt_r R l T L1 L2 -. - -(* Compact definition of llpx_sn_alt_r **************************************) - -lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,l. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 - ) → llpx_sn_alt_r R l T L1 L2. -#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/ -qed. - -lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv. - (∀L1,L2,T,l. |L1| = |L2| → ( - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2 - ) → S l T L1 L2) → - ∀L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → S l T L1 L2. -#R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l -#L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/ -qed-. - -lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2. -#R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l -#L1 #L2 #T #l #HL12 #IH @conj // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2 → - llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R l T L1 L2. -#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H -#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 -elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 // -/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/ -qed-. - -lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 → - llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). -#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H -#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 -[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 - /3 width=9 by nlift_bind_sn, and3_intro/ -| lapply (yle_inv_succ1 … Hli) -Hli * #Hli #Hi (lift_inv_sort1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l - /2 width=1 by llpx_sn_sort/ -| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H - * #Hli #H destruct - [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l - /2 width=1 by llpx_sn_skip/ - | elim (ylt_yle_false … Hil0) -L1 -L2 -K1 -K2 -m -Hil0 - /3 width=3 by yle_trans, yle_inj/ - ] -| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H - * #Hli #H destruct [ -HK12 | -IHK12 ] - [ elim (drop_trans_lt … HLK1 … HK11) // -K1 - elim (drop_trans_lt … HLK2 … HK22) // -Hli -K2 - /3 width=18 by llpx_sn_lref/ - | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1 - lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hli -Hl0 -K2 - /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/ - ] -| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H - * #Hil #H destruct - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12 - [ /3 width=7 by llpx_sn_free, drop_fwd_be/ - | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1 - lapply (drop_fwd_length … HLK2) -HLK2 #HLK2 - @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) - ] -| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l -m - /2 width=1 by llpx_sn_gref/ -| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H - #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/ -| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H - #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ -] -qed-. - -lemma llpx_sn_lift_ge: ∀R,K1,K2,T,l0. llpx_sn R l0 T K1 K2 → - ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀U. ⬆[l, m] T ≡ U → l ≤ l0 → llpx_sn R (l0+m) U L1 L2. -#R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0 -[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l - /2 width=1 by llpx_sn_sort/ -| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H - * #_ #H destruct - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 - [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/ - | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/ - ] -| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H - * #Hil #H destruct - [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -m -Hil0 - /3 width=3 by ylt_yle_trans, ylt_inj/ - | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1 - lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hil -Hl0 -K2 - /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/ - ] -| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H - * #Hil #H destruct - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12 - [ /3 width=7 by llpx_sn_free, drop_fwd_be/ - | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1 - lapply (drop_fwd_length … HLK2) -HLK2 #HLK2 - @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) - ] -| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l - /2 width=1 by llpx_sn_gref/ -| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H - #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_skip, yle_succ/ -| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H - #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/ -] -qed-. - -(* Inversion lemmas on relocation *******************************************) - -lemma llpx_sn_inv_lift_le: ∀R. d_deliftable_sn R → - ∀L1,L2,U,l0. llpx_sn R l0 U L1 L2 → - ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀T. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 T K1 K2. -#R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 -[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m - /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H - * #_ #H destruct - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 - [ /2 width=1 by llpx_sn_skip/ - | /3 width=3 by llpx_sn_skip, yle_ylt_trans/ - ] -| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H - * #Hil #H destruct [ -HK12 | -IHK12 ] - [ elim (drop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1 - elim (drop_conf_lt … HLK2 … HLK22) // -Hil -L2 #L2 #V2 #HKL2 #HKL22 #HVW2 - elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12 - lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct - /3 width=10 by llpx_sn_lref/ - | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 - lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 - elim (yle_inv_plus_inj2 … Hil) -Hil /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *) - ] -| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H - * #_ #H destruct - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) - [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 - lapply (drop_fwd_length_le4 … HLK2) -HLK2 - #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) - | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H - lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H - /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ - ] -| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m - /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 … H) -H - #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/ -| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat2 … H) -H - #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ -] -qed-. - -lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 → - ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀T. ⬆[l, m] T ≡ U → l ≤ l0 → l0 ≤ l + m → llpx_sn R l T K1 K2. -#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 -[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m - /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H - * #Hil #H destruct - [ lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 - -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/ - | elim (ylt_yle_false … Hil0) -L1 -L2 -Hl0 -Hil0 - /3 width=3 by yle_trans, yle_inj/ (**) (* slow *) - ] -| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H - * #Hil #H destruct - [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0m -Hil0 - /3 width=3 by ylt_yle_trans, ylt_inj/ - | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 - lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0m - elim (yle_inv_plus_inj2 … Hil) -Hil /3 width=9 by llpx_sn_lref, yle_inj/ - ] -| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H - * #_ #H destruct - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) - [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 - lapply (drop_fwd_length_le4 … HLK2) -HLK2 - #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) - | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H - lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H - /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ - ] -| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m - /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_bind2 … H) -H - #V #T #HVW #HTU #H destruct - @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) - @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/ -| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_flat2 … H) -H - #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ -] -qed-. - -lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 → - ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → - ∀T. ⬆[l, m] T ≡ U → l + m ≤ l0 → llpx_sn R (l0-m) T K1 K2. -#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 -[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l - /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H - * #Hil #H destruct [ -Hil0 | -Hlml0 ] - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 - [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/ - | elim (yle_inv_plus_inj2 … Hil) -Hil - /3 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx/ - ] -| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H - * #Hil #H destruct - [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0 - /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/ - | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 - lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hlml0 -Hil - /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/ - ] -| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H - * #_ #H destruct - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) - [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 - lapply (drop_fwd_length_le4 … HLK2) -HLK2 - #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) - | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H - lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H - /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ - ] -| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l - /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_bind2 … H) -H - #V #T #HVW #HTU #H destruct - @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) - yminus_Y_inj #K1 #HK12 #HLK1 - lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ -| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/ -| /4 width=1 by llpx_sn_bind, lreq_succ/ -] -qed-. - -lemma llpx_sn_lreq_trans: ∀R,L,L1,T,l. llpx_sn R l T L L1 → - ∀L2. L1 ⩬[l, ∞] L2 → llpx_sn R l T L L2. -#R #L #L1 #T #l #H elim H -L -L1 -T -l -/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/ -[ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12 - elim (lreq_drop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2 - lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ -| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/ -| /4 width=1 by llpx_sn_bind, lreq_succ/ -] -qed-. - -lemma llpx_sn_lreq_repl: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → ∀K1. K1 ⩬[l, ∞] L1 → - ∀K2. L2 ⩬[l, ∞] K2 → llpx_sn R l T K1 K2. -/3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-. - -lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) → - ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2). -#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT -/3 width=7 by llpx_sn_lreq_repl, lreq_succ/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_tc.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_tc.ma deleted file mode 100644 index 13fc14e82..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_tc.ma +++ /dev/null @@ -1,26 +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_2/multiple/llpx_sn_drop.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -(* Properties about transitive closure **************************************) - -lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) → - ∀I,L,V1,V2,T. LTC … R L V1 V2 → - LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2). -#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2 -/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma new file mode 100644 index 000000000..065135c9a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma @@ -0,0 +1,124 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/rdropstar_3.ma". +include "basic_2/notation/relations/rdropstar_4.ma". +include "basic_2/substitution/drop.ma". +include "basic_2/multiple/mr2_minus.ma". +include "basic_2/multiple/lifts_vector.ma". + +(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) + +inductive drops (s:bool): list2 ynat 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 ylt_to_minus/ #K #V pluss_SO2 >pluss_SO2 + >yminus_succ2 >ylt_inv_O1 /2 width=1 by ylt_to_minus/ commutative_plus (**) (* (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 *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_drop.ma new file mode 100644 index 000000000..2bf29c636 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_drop.ma @@ -0,0 +1,35 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/substitution/drop_drop.ma". +include "basic_2/multiple/drops.ma". + +(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) + +(* Properties concerning basic local environment slicing ********************) + +lemma drops_drop_trans: ∀L1,L,cs. ⬇*[Ⓕ, cs] L1 ≡ L → ∀L2,i. ⬇[i] L ≡ L2 → + ∃∃L0,cs0,i0. ⬇[i0] L1 ≡ L0 & ⬇*[Ⓕ, cs0] L0 ≡ L2 & + @⦃i, cs⦄ ≡ i0 & cs ▭ i ≡ cs0. +#L1 #L #cs #H elim H -L1 -L -cs +[ /2 width=7 by drops_nil, minuss_nil, at_nil, ex4_3_intro/ +| #L1 #L0 #L #cs #l #m #_ #HL0 #IHL0 #L2 #i #HL2 + elim (ylt_split i l) #Hil + [ elim (drop_trans_le … HL0 … HL2) -L /2 width=2 by ylt_fwd_le/ + #L #HL0 #HL2 elim (IHL0 … HL0) -L0 /3 width=7 by drops_cons, minuss_lt, at_lt, ex4_3_intro/ + | lapply (drop_trans_ge … HL0 … HL2 ?) -L // #HL02 + elim (IHL0 … HL02) -L0 /3 width=7 by minuss_ge, at_ge, ex4_3_intro/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_drops.ma new file mode 100644 index 000000000..c2b0b30d1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_drops.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/drops_drop.ma". + +(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************) + +(* Main properties **********************************************************) + +(* Basic_1: was: drop1_trans *) +theorem drops_trans: ∀L,L2,s,cs2. ⬇*[s, cs2] L ≡ L2 → ∀L1,cs1. ⬇*[s, cs1] L1 ≡ L → + ⬇*[s, cs2 @@ cs1] L1 ≡ L2. +#L #L2 #s #cs2 #H elim H -L -L2 -cs2 /3 width=3 by drops_cons/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fleq.ma new file mode 100644 index 000000000..1ca5528a6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fleq.ma @@ -0,0 +1,43 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/lazyeq_7.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR CLOSURES ********************************************) + +inductive fleq (l) (G) (L1) (T): relation3 genv lenv term ≝ +| fleq_intro: ∀L2. L1 ≡[T, l] L2 → fleq l G L1 T G L2 T +. + +interpretation + "lazy equivalence (closure)" + 'LazyEq l G1 L1 T1 G2 L2 T2 = (fleq l G1 L1 T1 G2 L2 T2). + +(* Basic properties *********************************************************) + +lemma fleq_refl: ∀l. tri_reflexive … (fleq l). +/2 width=1 by fleq_intro/ qed. + +lemma fleq_sym: ∀l. tri_symmetric … (fleq l). +#l #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma fleq_inv_gen: ∀G1,G2,L1,L2,T1,T2,l. ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄ → + ∧∧ G1 = G2 & L1 ≡[T1, l] L2 & T1 = T2. +#G1 #G2 #L1 #L2 #T1 #T2 #l * -G2 -L2 -T2 /2 width=1 by and3_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fleq_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fleq_fleq.ma new file mode 100644 index 000000000..915be044e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fleq_fleq.ma @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/lleq_lleq.ma". +include "basic_2/multiple/fleq.ma". + +(* LAZY EQUIVALENCE FOR CLOSURES *******************************************) + +(* Main properties **********************************************************) + +theorem fleq_trans: ∀l. tri_transitive … (fleq l). +#l #G1 #G #L1 #L #T1 #T * -G -L -T +#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2 +/3 width=3 by lleq_trans, fleq_intro/ +qed-. + +theorem fleq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2,l. + ⦃G, L, T⦄ ≡[l] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[l] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄. +/3 width=5 by fleq_trans, fleq_sym/ qed-. + +theorem fleq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T,l. + ⦃G1, L1, T1⦄ ≡[l] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[l] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄. +/3 width=5 by fleq_trans, fleq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqup.ma new file mode 100644 index 000000000..b126081ba --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqup.ma @@ -0,0 +1,109 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/suptermplus_6.ma". +include "basic_2/substitution/fqu.ma". + +(* PLUS-ITERATED SUPCLOSURE *************************************************) + +definition fqup: tri_relation genv lenv term ≝ tri_TC … fqu. + +interpretation "plus-iterated structural successor (closure)" + 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup G1 L1 T1 G2 L2 T2). + +(* Basic properties *********************************************************) + +lemma fqu_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +/2 width=1 by tri_inj/ qed. + +lemma fqup_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. + ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +/2 width=5 by tri_step/ qed. + +lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. + ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +/2 width=5 by tri_TC_strap/ qed. + +lemma fqup_drop: ∀G1,G2,L1,K1,K2,T1,T2,U1,m. ⬇[m] L1 ≡ K1 → ⬆[0, m] T1 ≡ U1 → + ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄. +#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #m #HLK1 #HTU1 #HT12 elim (eq_or_gt … m) #H destruct +[ >(drop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 // +| /3 width=5 by fqup_strap2, fqu_drop_lt/ +] +qed-. + +lemma fqup_lref: ∀I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄. +/3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_drop/ qed. + +lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+ ⦃G, L, V⦄. +/2 width=1 by fqu_pair_sn, fqu_fqup/ qed. + +lemma fqup_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊐+ ⦃G, L.ⓑ{I}V, T⦄. +/2 width=1 by fqu_bind_dx, fqu_fqup/ qed. + +lemma fqup_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+ ⦃G, L, T⦄. +/2 width=1 by fqu_flat_dx, fqu_fqup/ qed. + +lemma fqup_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+ ⦃G, L, V2⦄. +/2 width=5 by fqu_pair_sn, fqup_strap1/ qed. + +lemma fqup_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I1}V1, T⦄. +/2 width=5 by fqu_flat_dx, fqup_strap1/ qed. + +lemma fqup_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I2}V2, T⦄. +/2 width=5 by fqu_bind_dx, fqup_strap1/ qed. + +(* Basic eliminators ********************************************************) + +lemma fqup_ind: ∀G1,L1,T1. ∀R:relation3 …. + (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) → + (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) → + ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2. +#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H +@(tri_TC_ind … IH1 IH2 G2 L2 T2 H) +qed-. + +lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. + (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G1 L1 T1) → + (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) → + ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G1 L1 T1. +#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H +@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H) +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2. + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}. +#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 +/3 width=3 by fqu_fwd_fw, transitive_lt/ +qed-. + +(* Advanced eliminators *****************************************************) + +lemma fqup_wf_ind: ∀R:relation3 …. ( + ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) → + R G1 L1 T1 + ) → ∀G1,L1,T1. R G1 L1 T1. +#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/ +qed-. + +lemma fqup_wf_ind_eq: ∀R:relation3 …. ( + ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) → + ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2 + ) → ∀G1,L1,T1. R G1 L1 T1. +#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fqup_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqup_fqup.ma new file mode 100644 index 000000000..48420a7cf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqup_fqup.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/fqup.ma". + +(* PLUS-ITERATED SUPCLOSURE *************************************************) + +(* Main properties **********************************************************) + +theorem fqup_trans: tri_transitive … fqup. +/2 width=5 by tri_TC_transitive/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus.ma new file mode 100644 index 000000000..92679770e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus.ma @@ -0,0 +1,83 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/suptermstar_6.ma". +include "basic_2/substitution/fquq.ma". +include "basic_2/multiple/fqup.ma". + +(* STAR-ITERATED SUPCLOSURE *************************************************) + +definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq. + +interpretation "star-iterated structural successor (closure)" + 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2). + +(* Basic eliminators ********************************************************) + +lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 → + (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) → + ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2. +#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H +@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) // +qed-. + +lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 → + (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) → + ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1. +#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H +@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) // +qed-. + +(* Basic properties *********************************************************) + +lemma fqus_refl: tri_reflexive … fqus. +/2 width=1 by tri_inj/ qed. + +lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +/2 width=1 by tri_inj/ qed. + +lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +/2 width=5 by tri_step/ qed-. + +lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +/2 width=5 by tri_TC_strap/ qed-. + +lemma fqus_drop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ → + ∀L1,U1,m. ⬇[m] L1 ≡ K1 → ⬆[0, m] T1 ≡ U1 → + ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄. +#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2 +/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/ +qed-. + +lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 +/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/ +qed. + +(* Basic forward lemmas *****************************************************) + +lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}. +#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2 +/3 width=3 by fquq_fwd_fw, transitive_le/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → + ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄. +#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/ +#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus_alt.ma new file mode 100644 index 000000000..c9ec457b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus_alt.ma @@ -0,0 +1,61 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/substitution/fquq_alt.ma". +include "basic_2/multiple/fqus.ma". + +(* STAR-ITERATED SUPCLOSURE *************************************************) + +(* Advanced inversion lemmas ************************************************) + +lemma fqus_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2). +#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 // +#G #G2 #L #L2 #T #T2 #_ #H2 * elim (fquq_inv_gen … H2) -H2 +[ /3 width=5 by fqup_strap1, or_introl/ +| * #HG #HL #HT destruct /2 width=1 by or_introl/ +| #H2 * #HG #HL #HT destruct /3 width=1 by fqu_fqup, or_introl/ +| * #H1G #H1L #H1T * #H2G #H2L #H2T destruct /2 width=1 by or_intror/ +] +qed-. + +(* Advanced properties ******************************************************) + +lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H1) -H1 +[ /2 width=5 by fqup_strap1/ +| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/ +] +qed-. + +lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H2) -H2 +[ /2 width=5 by fqup_strap2/ +| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/ +] +qed-. + +lemma fqus_fqup_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2 +/2 width=5 by fqus_strap1_fqu, fqup_strap1/ +qed-. + +lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → + ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄. +#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1 +/3 width=5 by fqus_strap2_fqu, fqup_strap2/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus_fqus.ma new file mode 100644 index 000000000..ef9902931 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqus_fqus.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/fqus.ma". + +(* STAR-ITERATED SUPCLOSURE *************************************************) + +(* Main properties **********************************************************) + +theorem fqus_trans: tri_transitive … fqus. +/2 width=5 by tri_TC_transitive/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/frees.ma new file mode 100644 index 000000000..15117986b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/frees.ma @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/freestar_3.ma". +include "basic_2/grammar/trace_sor.ma". +include "basic_2/grammar/lenv.ma". + +(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) + +inductive frees: relation3 lenv term trace ≝ +| frees_atom: ∀I. frees (⋆) (⓪{I}) (◊) +| frees_sort: ∀L,k,cs. frees L (⋆k) cs → + ∀I,T. frees (L.ⓑ{I}T) (⋆k) (Ⓕ @ cs) +| frees_zero: ∀L,T,cs. frees L T cs → + ∀I. frees (L.ⓑ{I}T) (#0) (Ⓣ @ cs) +| frees_lref: ∀L,i,cs. frees L (#i) cs → + ∀I,T. frees (L.ⓑ{I}T) (#(S i)) (Ⓕ @ cs) +| frees_gref: ∀L,p,cs. frees L (§p) cs → + ∀I,T. frees (L.ⓑ{I}T) (§p) (Ⓕ @ cs) +| frees_bind: ∀cv,ct,cs. cv ⋓ ct ≡ cs → + ∀L,V. frees L V cv → ∀I,T,b. frees (L.ⓑ{I}V) T (b @ ct) → + ∀a. frees L (ⓑ{a,I}V.T) cs +| frees_flat: ∀cv,ct,cs. cv ⋓ ct ≡ cs → + ∀L,V. frees L V cv → ∀T. frees L T ct → + ∀I. frees L (ⓕ{I}V.T) cs +. + +interpretation + "context-sensitive free variables (term)" + 'FreeStar L T cs = (frees L T cs). 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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma new file mode 100644 index 000000000..b0e32002c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts.ma @@ -0,0 +1,101 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/relocation/lifts.ma". + +(* GENERIC RELOCATION *******************************************************) + +(* Main properties **********************************************************) + +(* Basic_2A1: includes: lift_inj *) +theorem lifts_inj: ∀t,T1,U. ⬆*[t] T1 ≡ U → ∀T2. ⬆*[t] T2 ≡ U → T1 = T2. +#t #T1 #U #H elim H -t -T1 -U +[ /2 width=2 by lifts_inv_sort2/ +| #i1 #j #t #Hi1j #X #HX elim (lifts_inv_lref2 … HX) -HX + /4 width=4 by at_inj, eq_f/ +| /2 width=2 by lifts_inv_gref2/ +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_bind2 … HX) -HX + #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/ +| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_flat2 … HX) -HX + #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/ +] +qed-. + +(* Basic_1: includes: lift_gen_lift *) +(* Basic_2A1: includes: lift_div_le lift_div_be *) +theorem lifts_div: ∀T,T2,t2. ⬆*[t2] T2 ≡ T → ∀T1,t. ⬆*[t] T1 ≡ T → + ∀t1. t2 ⊚ t1 ≡ t → ⬆*[t1] T1 ≡ T2. +#T #T2 #t2 #H elim H -T -T2 -t2 +[ #k #t2 #T1 #t #H >(lifts_inv_sort2 … H) -T1 // +| #i2 #i #t2 #Hi2 #T1 #t #H #t1 #Ht21 elim (lifts_inv_lref2 … H) -H + #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/ +| #p #t2 #T1 #t #H >(lifts_inv_gref2 … H) -T1 // +| #a #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #H + elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct + /4 width=3 by lifts_bind, after_true/ +| #I #W2 #W #U2 #U #t2 #_ #_ #IHW #IHU #T1 #t #H + elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct + /3 width=3 by lifts_flat/ +] +qed-. + +(* Basic_2A1: includes: lift_mono *) +theorem lifts_mono: ∀t,T,U1. ⬆*[t] T ≡ U1 → ∀U2. ⬆*[t] T ≡ U2 → U1 = U2. +#t #T #U1 #H elim H -t -T -U1 +[ /2 width=2 by lifts_inv_sort1/ +| #i1 #j #t #Hi1j #X #HX elim (lifts_inv_lref1 … HX) -HX + /4 width=4 by at_mono, eq_f/ +| /2 width=2 by lifts_inv_gref1/ +| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_bind1 … HX) -HX + #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/ +| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV12 #IHT12 #X #HX elim (lifts_inv_flat1 … HX) -HX + #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/ +] +qed-. + +(* Basic_1: was: lift1_lift1 (left to right) *) +(* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *) +(* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *) +theorem lifts_trans: ∀T1,T,t1. ⬆*[t1] T1 ≡ T → ∀T2,t2. ⬆*[t2] T ≡ T2 → + ∀t. t2 ⊚ t1 ≡ t → ⬆*[t] T1 ≡ T2. +#T1 #T #t1 #H elim H -T1 -T -t1 +[ #k #t1 #T2 #t2 #H >(lifts_inv_sort1 … H) -T2 // +| #i1 #i #t1 #Hi1 #T2 #t2 #H #t #Ht21 elim (lifts_inv_lref1 … H) -H + #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/ +| #p #t1 #T2 #t2 #H >(lifts_inv_gref1 … H) -T2 // +| #a #I #W1 #W #U1 #U #t1 #_ #_ #IHW #IHU #T2 #t2 #H + elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct + /4 width=3 by lifts_bind, after_true/ +| #I #W1 #W #U1 #U #t1 #_ #_ #IHW #IHU #T2 #t2 #H + elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct + /3 width=3 by lifts_flat/ +] +qed-. + +(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *) +theorem lifts_conf: ∀T,T1,t1. ⬆*[t1] T ≡ T1 → ∀T2,t. ⬆*[t] T ≡ T2 → + ∀t2. t2 ⊚ t1 ≡ t → ⬆*[t2] T1 ≡ T2. +#T #T1 #t1 #H elim H -T -T1 -t1 +[ #k #t1 #T2 #t #H >(lifts_inv_sort1 … H) -T2 // +| #i #i1 #t1 #Hi1 #T2 #t #H #t2 #Ht21 elim (lifts_inv_lref1 … H) -H + #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/ +| #p #t1 #T2 #t #H >(lifts_inv_gref1 … H) -T2 // +| #a #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #H + elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct + /4 width=3 by lifts_bind, after_true/ +| #I #W #W1 #U #U1 #t1 #_ #_ #IHW #IHU #T2 #t #H + elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct + /3 width=3 by lifts_flat/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts_vector.ma new file mode 100644 index 000000000..fb5953c26 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_lifts_vector.ma @@ -0,0 +1,53 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/relocation/lifts_lifts.ma". +include "basic_2/relocation/lifts_vector.ma". + +(* GENERIC TERM VECTOR RELOCATION *******************************************) + +(* Main properties **********************************************************) + +(* Basic_1: includes: lifts_inj *) +theorem liftsv_inj: ∀T1s,Us,t. ⬆*[t] T1s ≡ Us → + ∀T2s. ⬆*[t] T2s ≡ Us → T1s = T2s. +#T1s #Us #t #H elim H -T1s -Us +[ #T2s #H >(liftsv_inv_nil2 … H) -H // +| #T1s #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct + elim (liftsv_inv_cons2 … H) -H #T2 #T2s #HT2U #HT2Us #H destruct + >(lifts_inj … HT1U … HT2U) -U /3 width=1 by eq_f/ +] +qed-. + +(* Basic_2A1: includes: liftv_mono *) +theorem liftsv_mono: ∀Ts,U1s,t. ⬆*[t] Ts ≡ U1s → + ∀U2s. ⬆*[t] Ts ≡ U2s → U1s = U2s. +#Ts #U1s #t #H elim H -Ts -U1s +[ #U2s #H >(liftsv_inv_nil1 … H) -H // +| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct + elim (liftsv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct + >(lifts_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/ +] +qed-. + +(* Basic_1: includes: lifts1_xhg (right to left) *) +(* Basic_2A1: includes: liftsv_liftv_trans_le *) +theorem liftsv_trans: ∀T1s,Ts,t1. ⬆*[t1] T1s ≡ Ts → ∀T2s,t2. ⬆*[t2] Ts ≡ T2s → + ∀t. t2 ⊚ t1 ≡ t → ⬆*[t] T1s ≡ T2s. +#T1s #Ts #t1 #H elim H -T1s -Ts +[ #T2s #t2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/ +| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #t2 #H elim (liftsv_inv_cons1 … H) -H + #T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_simple.ma new file mode 100644 index 000000000..389f724d2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_simple.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/grammar/term_simple.ma". +include "basic_2/relocation/lifts.ma". + +(* GENERIC TERM RELOCATION **************************************************) + +(* Forward lemmas on simple terms *******************************************) + +(* Basic_2A1: includes: lift_simple_dx *) +lemma lifts_simple_dx: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄. +#T1 #T2 #t #H elim H -T1 -T2 -t // +#a #I #V1 #V2 #T1 #T2 #t #_ #_ #_ #_ #H elim (simple_inv_bind … H) +qed-. + +(* Basic_2A1: includes: lift_simple_sn *) +lemma lifts_simple_sn: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄. +#T1 #T2 #t #H elim H -T1 -T2 -t // +#a #I #V1 #V2 #T1 #T2 #t #_ #_ #_ #_ #H elim (simple_inv_bind … H) +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma new file mode 100644 index 000000000..89ead4cf5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_vector.ma @@ -0,0 +1,114 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/grammar/term_vector.ma". +include "basic_2/relocation/lifts.ma". + +(* GENERIC TERM VECTOR RELOCATION *******************************************) + +(* Basic_2A1: includes: liftv_nil liftv_cons *) +inductive liftsv (t:trace) : relation (list term) ≝ +| liftsv_nil : liftsv t (◊) (◊) +| liftsv_cons: ∀T1s,T2s,T1,T2. + ⬆*[t] T1 ≡ T2 → liftsv t T1s T2s → + liftsv t (T1 @ T1s) (T2 @ T2s) +. + +interpretation "generic relocation (vector)" + 'RLiftStar t T1s T2s = (liftsv t T1s T2s). + +(* Basic inversion lemmas ***************************************************) + +fact liftsv_inv_nil1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → X = ◊ → Y = ◊. +#X #Y #t * -X -Y // +#T1s #T2s #T1 #T2 #_ #_ #H destruct +qed-. + +(* Basic_2A1: includes: liftv_inv_nil1 *) +lemma liftsv_inv_nil1: ∀Y,t. ⬆*[t] ◊ ≡ Y → Y = ◊. +/2 width=5 by liftsv_inv_nil1_aux/ qed-. + +fact liftsv_inv_cons1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → + ∀T1,T1s. X = T1 @ T1s → + ∃∃T2,T2s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s & + Y = T2 @ T2s. +#X #Y #t * -X -Y +[ #U1 #U1s #H destruct +| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +(* Basic_2A1: includes: liftv_inv_cons1 *) +lemma liftsv_inv_cons1: ∀T1,T1s,Y,t. ⬆*[t] T1 @ T1s ≡ Y → + ∃∃T2,T2s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s & + Y = T2 @ T2s. +/2 width=3 by liftsv_inv_cons1_aux/ qed-. + +fact liftsv_inv_nil2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → Y = ◊ → X = ◊. +#X #Y #t * -X -Y // +#T1s #T2s #T1 #T2 #_ #_ #H destruct +qed-. + +lemma liftsv_inv_nil2: ∀X,t. ⬆*[t] X ≡ ◊ → X = ◊. +/2 width=5 by liftsv_inv_nil2_aux/ qed-. + +fact liftsv_inv_cons2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → + ∀T2,T2s. Y = T2 @ T2s → + ∃∃T1,T1s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s & + X = T1 @ T1s. +#X #Y #t * -X -Y +[ #U2 #U2s #H destruct +| #T1s #T2s #T1 #T2 #HT12 #HT12s #U2 #U2s #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma liftsv_inv_cons2: ∀X,T2,T2s,t. ⬆*[t] X ≡ T2 @ T2s → + ∃∃T1,T1s. ⬆*[t] T1 ≡ T2 & ⬆*[t] T1s ≡ T2s & + X = T1 @ T1s. +/2 width=3 by liftsv_inv_cons2_aux/ qed-. + +(* Basic_1: was: lifts1_flat (left to right) *) +lemma lifts_inv_applv1: ∀V1s,U1,T2,t. ⬆*[t] Ⓐ V1s.U1 ≡ T2 → + ∃∃V2s,U2. ⬆*[t] V1s ≡ V2s & ⬆*[t] U1 ≡ U2 & + T2 = Ⓐ V2s.U2. +#V1s elim V1s -V1s +[ /3 width=5 by ex3_2_intro, liftsv_nil/ +| #V1 #V1s #IHV1s #T1 #X #t #H elim (lifts_inv_flat1 … H) -H + #V2 #Y #HV12 #HY #H destruct elim (IHV1s … HY) -IHV1s -HY + #V2s #T2 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/ +] +qed-. + +lemma lifts_inv_applv2: ∀V2s,U2,T1,t. ⬆*[t] T1 ≡ Ⓐ V2s.U2 → + ∃∃V1s,U1. ⬆*[t] V1s ≡ V2s & ⬆*[t] U1 ≡ U2 & + T1 = Ⓐ V1s.U1. +#V2s elim V2s -V2s +[ /3 width=5 by ex3_2_intro, liftsv_nil/ +| #V2 #V2s #IHV2s #T2 #X #t #H elim (lifts_inv_flat2 … H) -H + #V1 #Y #HV12 #HY #H destruct elim (IHV2s … HY) -IHV2s -HY + #V1s #T1 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/ +] +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: lifts1_flat (right to left) *) +lemma lifts_applv: ∀V1s,V2s,t. ⬆*[t] V1s ≡ V2s → + ∀T1,T2. ⬆*[t] T1 ≡ T2 → + ⬆*[t] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2. +#V1s #V2s #t #H elim H -V1s -V2s /3 width=1 by lifts_flat/ +qed. + +(* Basic_2A1: removed theorems 1: liftv_total *) +(* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_weight.ma new file mode 100644 index 000000000..17952bba8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts_weight.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/grammar/term_weight.ma". +include "basic_2/relocation/lifts.ma". + +(* GENERIC TERM RELOCATION **************************************************) + +(* Forward lemmas on weight for terms ***************************************) + +(* Basic_2A1: includes: lift_fwd_tw *) +lemma lifts_fwd_tw: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → ♯{T1} = ♯{T2}. +#T1 #T2 #t #H elim H -T1 -T2 -t normalize // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma new file mode 100644 index 000000000..1f57f73be --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma @@ -0,0 +1,160 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/notation/relations/lazyeq_4.ma". +include "basic_2/multiple/llpx_sn.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2. + +definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq. + +interpretation + "lazy equivalence (local environment)" + 'LazyEq T l L1 L2 = (lleq l T L1 L2). + +definition lleq_transitive: predicate (relation3 lenv term term) ≝ + λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2. + +(* Basic inversion lemmas ***************************************************) + +lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. ( + ∀L1,L2,l,k. |L1| = |L2| → R l (⋆k) L1 L2 + ) → ( + ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2 + ) → ( + ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → + ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2 + ) → ( + ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2 + ) → ( + ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2 + ) → ( + ∀a,I,L1,L2,V,T,l. + L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → + R l V L1 L2 → R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2 + ) → ( + ∀I,L1,L2,V,T,l. + L1 ≡[V, l]L2 → L1 ≡[T, l] L2 → + R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2 + ) → + ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2. +#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/ +qed-. + +lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,l. L1 ≡[ⓑ{a,I}V.T, l] L2 → + L1 ≡[V, l] L2 ∧ L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_inv_bind/ qed-. + +lemma lleq_inv_flat: ∀I,L1,L2,V,T,l. L1 ≡[ⓕ{I}V.T, l] L2 → + L1 ≡[V, l] L2 ∧ L1 ≡[T, l] L2. +/2 width=2 by llpx_sn_inv_flat/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lleq_fwd_length: ∀L1,L2,T,l. L1 ≡[T, l] L2 → |L1| = |L2|. +/2 width=4 by llpx_sn_fwd_length/ qed-. + +lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → + ∨∨ |L1| ≤ i ∧ |L2| ≤ i + | yinj i < l + | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V & + ⬇[i] L2 ≡ K2.ⓑ{I}V & + K1 ≡[V, yinj 0] K2 & l ≤ yinj i. +#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/ +* /3 width=7 by or3_intro2, ex4_4_intro/ +qed-. + +lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 → + ∃K2. ⬇[i] L2 ≡ K2. +/2 width=7 by llpx_sn_fwd_drop_sn/ qed-. + +lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 → + ∃K1. ⬇[i] L1 ≡ K1. +/2 width=7 by llpx_sn_fwd_drop_dx/ qed-. + +lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,l. + L1 ≡[ⓑ{a,I}V.T, l] L2 → L1 ≡[V, l] L2. +/2 width=4 by llpx_sn_fwd_bind_sn/ qed-. + +lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,l. + L1 ≡[ⓑ{a,I}V.T, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_fwd_bind_dx/ qed-. + +lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,l. + L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[V, l] L2. +/2 width=3 by llpx_sn_fwd_flat_sn/ qed-. + +lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,l. + L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[T, l] L2. +/2 width=3 by llpx_sn_fwd_flat_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma lleq_sort: ∀L1,L2,l,k. |L1| = |L2| → L1 ≡[⋆k, l] L2. +/2 width=1 by llpx_sn_sort/ qed. + +lemma lleq_skip: ∀L1,L2,l,i. yinj i < l → |L1| = |L2| → L1 ≡[#i, l] L2. +/2 width=1 by llpx_sn_skip/ qed. + +lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → + ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2. +/2 width=9 by llpx_sn_lref/ qed. + +lemma lleq_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, l] L2. +/2 width=1 by llpx_sn_free/ qed. + +lemma lleq_gref: ∀L1,L2,l,p. |L1| = |L2| → L1 ≡[§p, l] L2. +/2 width=1 by llpx_sn_gref/ qed. + +lemma lleq_bind: ∀a,I,L1,L2,V,T,l. + L1 ≡[V, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → + L1 ≡[ⓑ{a,I}V.T, l] L2. +/2 width=1 by llpx_sn_bind/ qed. + +lemma lleq_flat: ∀I,L1,L2,V,T,l. + L1 ≡[V, l] L2 → L1 ≡[T, l] L2 → L1 ≡[ⓕ{I}V.T, l] L2. +/2 width=1 by llpx_sn_flat/ qed. + +lemma lleq_refl: ∀l,T. reflexive … (lleq l T). +/2 width=1 by llpx_sn_refl/ qed. + +lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2. +/2 width=1 by llpx_sn_Y/ qed. + +lemma lleq_sym: ∀l,T. symmetric … (lleq l T). +#l #T #L1 #L2 #H @(lleq_ind … H) -l -T -L1 -L2 +/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/ +qed-. + +lemma lleq_ge_up: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀T,l,m. ⬆[l, m] T ≡ U → + lt ≤ l + m → L1 ≡[U, l] L2. +/2 width=6 by llpx_sn_ge_up/ qed-. + +lemma lleq_ge: ∀L1,L2,T,l1. L1 ≡[T, l1] L2 → ∀l2. l1 ≤ l2 → L1 ≡[T, l2] L2. +/2 width=3 by llpx_sn_ge/ qed-. + +lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → + L1 ≡[ⓑ{a,I}V.T, 0] L2. +/2 width=1 by llpx_sn_bind_O/ qed-. + +(* Advanceded properties on lazy pointwise extensions ************************) + +lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → llpx_sn R l T L1 L2. +/2 width=3 by llpx_sn_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_alt.ma new file mode 100644 index 000000000..d2919f5b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_alt.ma @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/llpx_sn_alt.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Alternative definition (not recursive) ***********************************) + +theorem lleq_intro_alt: ∀L1,L2,T,l. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ V1 = V2 + ) → L1 ≡[T, l] L2. +#L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // +qed. + +theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ V1 = V2. +#L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H +#HL12 #IH @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_alt_rec.ma new file mode 100644 index 000000000..440e0f510 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_alt_rec.ma @@ -0,0 +1,54 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/llpx_sn_alt_rec.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Alternative definition (recursive) ***************************************) + +theorem lleq_intro_alt_r: ∀L1,L2,T,l. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 + ) → L1 ≡[T, l] L2. +#L1 #L2 #T #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ +qed. + +theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv. + (∀L1,L2,T,l. |L1| = |L2| → ( + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2 + ) → S l T L1 L2) → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → S l T L1 L2. +#S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -l +#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/ +qed-. + +theorem lleq_inv_alt_r: ∀L1,L2,T,l. L1 ≡[T, l] L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2. +#L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r … H) -H +#HL12 #IH @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_drop.ma new file mode 100644 index 000000000..09ee8a495 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_drop.ma @@ -0,0 +1,150 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/llpx_sn_drop.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Advanced properties ******************************************************) + +lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → + ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W. +/2 width=7 by llpx_sn_bind_repl_O/ qed-. + +lemma lleq_dec: ∀T,L1,L2,l. Decidable (L1 ≡[T, l] L2). +/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-. + +lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → + ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L. +#R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l +[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/ +|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/ +| #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2 + /3 width=11 by llpx_sn_lref/ +| #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H + /3 width=1 by llpx_sn_bind/ +| #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H + /3 width=1 by llpx_sn_flat/ +] +qed-. + +lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → + ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L. +/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → + ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2. +#L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2 +/2 width=3 by ex2_intro/ +qed-. + +lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → + ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2. +#L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1 +/2 width=3 by ex2_intro/ +qed-. + +lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I1,I2,K1,K2,V1,V2. + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2. +/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. + +lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, 0] K2. +#L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2 +elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) // +qed-. + +lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 → + ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, 0] K2 → L1 ≡[T, l] L2. +/2 width=9 by llpx_sn_inv_S/ qed-. + +lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → + L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_inv_bind_O/ qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → + ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → + i < l ∨ + ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. +#L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 +[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ +qed-. + +lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → + ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → + i < l ∨ + ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. +#L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 +[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ +qed-. + +lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → + L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-. + +(* Properties on relocation *************************************************) + +lemma lleq_lift_le: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → + ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀U. ⬆[l, m] T ≡ U → lt ≤ l → L1 ≡[U, lt] L2. +/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-. + +lemma lleq_lift_ge: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → + ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀U. ⬆[l, m] T ≡ U → l ≤ lt → L1 ≡[U, lt+m] L2. +/2 width=9 by llpx_sn_lift_ge/ qed-. + +(* Inversion lemmas on relocation *******************************************) + +lemma lleq_inv_lift_le: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → lt ≤ l → K1 ≡[T, lt] K2. +/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-. + +lemma lleq_inv_lift_be: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l ≤ lt → lt ≤ l + m → K1 ≡[T, l] K2. +/2 width=11 by llpx_sn_inv_lift_be/ qed-. + +lemma lleq_inv_lift_ge: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l + m ≤ lt → K1 ≡[T, lt-m] K2. +/2 width=9 by llpx_sn_inv_lift_ge/ qed-. + +(* Inversion lemmas on negated lazy quivalence for local environments *******) + +lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) → + (L1 ≡[V, l] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → ⊥). +/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-. + +lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) → + (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥). +/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-. + +lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) → + (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥). +/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_fqus.ma new file mode 100644 index 000000000..28bb175ed --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_fqus.ma @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/fqus_alt.ma". +include "basic_2/multiple/lleq_drop.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Properties on supclosure *************************************************) + +lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U +[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H // + #K1 #H1 #H2 lapply (drop_inv_O2 … H1) -H1 + #H destruct /2 width=3 by fqu_lref_O, ex2_intro/ +| * [ #a ] #I #G #L2 #V #T #L1 #H + [ elim (lleq_inv_bind … H) + | elim (lleq_inv_flat … H) + ] -H + /2 width=3 by fqu_pair_sn, ex2_intro/ +| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H + #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/ +| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H + /2 width=3 by fqu_flat_dx, ex2_intro/ +| #G #L2 #K2 #T #U #m #HLK2 #HTU #L1 #HL12 + elim (drop_O1_le (Ⓕ) (m+1) L1) + [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/ + | lapply (drop_fwd_length_le2 … HLK2) -K2 + lapply (lleq_fwd_length … HL12) -T -U // + ] +] +qed-. + +lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H +[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/ +| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ +] +qed-. + +lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U +[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2 + /3 width=3 by fqu_fqup, ex2_intro/ +| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2 + #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K + /3 width=5 by fqup_strap1, ex2_intro/ +] +qed-. + +lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H +[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/ +| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lleq.ma new file mode 100644 index 000000000..630b41a32 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lleq.ma @@ -0,0 +1,39 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/lleq_drop.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Main properties **********************************************************) + +theorem lleq_trans: ∀l,T. Transitive … (lleq l T). +/2 width=3 by lleq_llpx_sn_trans/ qed-. + +theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2. +/3 width=3 by lleq_trans, lleq_sym/ qed-. + +theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2. +/3 width=3 by lleq_trans, lleq_sym/ qed-. + +(* Advanced properies on negated lazy equivalence *****************************) + +(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *) +lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L → + ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥). +/3 width=3 by lleq_canc_sn/ qed-. + +lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L → + ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥). +/3 width=3 by lleq_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_llor.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_llor.ma new file mode 100644 index 000000000..06893aad6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_llor.ma @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/llor.ma". +include "basic_2/multiple/llpx_sn_frees.ma". +include "basic_2/multiple/lleq_alt.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Properties on pointwise union for local environments **********************) + +lemma llpx_sn_llor_dx: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → + ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L2 ≡[T, l] L. +#R #H1R #H2R #L1 #L2 #T #l #H1 #L #H2 +lapply (llpx_sn_frees_trans … H1R H2R … H1) -H1R -H2R #HR +elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1 +elim H2 -H2 #_ #HL1 #IH2 +@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK +lapply (drop_fwd_length_lt2 … HLK) #HiL +elim (drop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1 +elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct +elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H +[ elim (ylt_yle_false … H) -H // +| elim H -H /2 width=1 by/ +] +qed. + +lemma llpx_sn_llor_dx_sym: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → + ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L ≡[T, l] L2. +/3 width=6 by llpx_sn_llor_dx, lleq_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lreq.ma new file mode 100644 index 000000000..8fa21a5da --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lreq.ma @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/llpx_sn_lreq.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Properties on equivalence for local environments *************************) + +lemma lreq_lleq_trans: ∀L2,L,T,l. L2 ≡[T, l] L → + ∀L1. L1 ⩬[l, ∞] L2 → L1 ≡[T, l] L. +/2 width=3 by lreq_llpx_sn_trans/ qed-. + +lemma lleq_lreq_trans: ∀L,L1,T,l. L ≡[T, l] L1 → + ∀L2. L1 ⩬[l, ∞] L2 → L ≡[T, l] L2. +/2 width=3 by llpx_sn_lreq_trans/ qed-. + +lemma lleq_lreq_repl: ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀K1. K1 ⩬[l, ∞] L1 → + ∀K2. L2 ⩬[l, ∞] K2 → K1 ≡[T, l] K2. +/2 width=5 by llpx_sn_lreq_repl/ qed-. + +lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 → + ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2. +/2 width=5 by llpx_sn_bind_repl_SO/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn.ma new file mode 100644 index 000000000..cf6e4ad76 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn.ma @@ -0,0 +1,209 @@ +(**************************************************************************) +(* ___ *) +(* ||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/ynat/ynat_plus.ma". +include "basic_2/substitution/drop.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝ +| llpx_sn_sort: ∀L1,L2,l,k. |L1| = |L2| → llpx_sn R l (⋆k) L1 L2 +| llpx_sn_skip: ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → llpx_sn R l (#i) L1 L2 +| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i → + ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 → + llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2 +| llpx_sn_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R l (#i) L1 L2 +| llpx_sn_gref: ∀L1,L2,l,p. |L1| = |L2| → llpx_sn R l (§p) L1 L2 +| llpx_sn_bind: ∀a,I,L1,L2,V,T,l. + llpx_sn R l V L1 L2 → llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → + llpx_sn R l (ⓑ{a,I}V.T) L1 L2 +| llpx_sn_flat: ∀I,L1,L2,V,T,l. + llpx_sn R l V L1 L2 → llpx_sn R l T L1 L2 → llpx_sn R l (ⓕ{I}V.T) L1 L2 +. + +(* Basic inversion lemmas ***************************************************) + +fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → + ∀a,I,V,T. X = ⓑ{a,I}V.T → + llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). +#R #L1 #L2 #X #l * -L1 -L2 -X -l +[ #L1 #L2 #l #k #_ #b #J #W #U #H destruct +| #L1 #L2 #l #i #_ #_ #b #J #W #U #H destruct +| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct +| #L1 #L2 #l #i #_ #_ #_ #b #J #W #U #H destruct +| #L1 #L2 #l #p #_ #b #J #W #U #H destruct +| #a #I #L1 #L2 #V #T #l #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/ +| #I #L1 #L2 #V #T #l #_ #_ #b #J #W #U #H destruct +] +qed-. + +lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → + llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). +/2 width=4 by llpx_sn_inv_bind_aux/ qed-. + +fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → + ∀I,V,T. X = ⓕ{I}V.T → + llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2. +#R #L1 #L2 #X #l * -L1 -L2 -X -l +[ #L1 #L2 #l #k #_ #J #W #U #H destruct +| #L1 #L2 #l #i #_ #_ #J #W #U #H destruct +| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #J #W #U #H destruct +| #L1 #L2 #l #i #_ #_ #_ #J #W #U #H destruct +| #L1 #L2 #l #p #_ #J #W #U #H destruct +| #a #I #L1 #L2 #V #T #l #_ #_ #J #W #U #H destruct +| #I #L1 #L2 #V #T #l #HV #HT #J #W #U #H destruct /2 width=1 by conj/ +] +qed-. + +lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 → + llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2. +/2 width=4 by llpx_sn_inv_flat_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma llpx_sn_fwd_length: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → |L1| = |L2|. +#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l // +#I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #HLK1 #HLK2 #_ #_ #HK12 +lapply (drop_fwd_length … HLK1) -HLK1 +lapply (drop_fwd_length … HLK2) -HLK2 +normalize // +qed-. + +lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → + ∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2. +#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H +#HL12 lapply (drop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by drop_O1_le/ +qed-. + +lemma llpx_sn_fwd_drop_dx: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → + ∀K2,i. ⬇[i] L2 ≡ K2 → ∃K1. ⬇[i] L1 ≡ K1. +#R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H +#HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/ +qed-. + +fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → ∀i. X = #i → + ∨∨ |L1| ≤ i ∧ |L2| ≤ i + | yinj i < l + | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & + ⬇[i] L2 ≡ K2.ⓑ{I}V2 & + llpx_sn R (yinj 0) V1 K1 K2 & + R K1 V1 V2 & l ≤ yinj i. +#R #L1 #L2 #X #l * -L1 -L2 -X -l +[ #L1 #L2 #l #k #_ #j #H destruct +| #L1 #L2 #l #i #_ #Hil #j #H destruct /2 width=1 by or3_intro1/ +| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 #j #H destruct + /3 width=9 by or3_intro2, ex5_5_intro/ +| #L1 #L2 #l #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/ +| #L1 #L2 #l #p #_ #j #H destruct +| #a #I #L1 #L2 #V #T #l #_ #_ #j #H destruct +| #I #L1 #L2 #V #T #l #_ #_ #j #H destruct +] +qed-. + +lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → + ∨∨ |L1| ≤ i ∧ |L2| ≤ i + | yinj i < l + | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & + ⬇[i] L2 ≡ K2.ⓑ{I}V2 & + llpx_sn R (yinj 0) V1 K1 K2 & + R K1 V1 V2 & l ≤ yinj i. +/2 width=3 by llpx_sn_fwd_lref_aux/ qed-. + +lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → + llpx_sn R l V L1 L2. +#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H // +qed-. + +lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → + llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). +#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H // +qed-. + +lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 → + llpx_sn R l V L1 L2. +#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H // +qed-. + +lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 → + llpx_sn R l T L1 L2. +#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H // +qed-. + +lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (②{I}V.T) L1 L2 → + llpx_sn R l V L1 L2. +#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/ +qed-. + +(* Basic properties *********************************************************) + +lemma llpx_sn_refl: ∀R. (∀L. reflexive … (R L)) → ∀T,L,l. llpx_sn R l T L L. +#R #HR #T #L @(f2_ind … rfw … L T) -L -T +#x #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/ +#i #Hx elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/ +#HiL #l elim (ylt_split i l) /2 width=1 by llpx_sn_skip/ +elim (drop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, drop_fwd_rfw/ +qed-. + +lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2. +#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T +#x #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/ +#a #I #V1 #T1 #Hx #L2 #HL12 +@llpx_sn_bind /2 width=1 by/ (**) (* explicit constructor *) +@IH -IH // normalize /2 width=1 by eq_f2/ +qed-. + +lemma llpx_sn_ge_up: ∀R,L1,L2,U,lt. llpx_sn R lt U L1 L2 → ∀T,l,m. ⬆[l, m] T ≡ U → + lt ≤ l + m → llpx_sn R l U L1 L2. +#R #L1 #L2 #U #lt #H elim H -L1 -L2 -U -lt +[ #L1 #L2 #lt #k #HL12 #X #l #m #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/ +| #L1 #L2 #lt #i #HL12 #Hilt #X #l #m #H #Hltlm + elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12 + elim (ylt_yle_false … Hilt) -Hilt + @(yle_trans … Hltlm) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *) +| #I #L1 #L2 #K1 #K2 #W1 #W2 #lt #i #Hlti #HLK1 #HLK2 #HW1 #HW12 #_ #X #l #m #H #_ + elim (lift_inv_lref2 … H) -H * #Hil #H destruct + [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12 + lapply (drop_fwd_length … HLK1) lapply (drop_fwd_length … HLK2) + normalize in ⊢ (%→%→?); -I -W1 -W2 -lt /3 width=1 by llpx_sn_skip, ylt_inj/ + | /3 width=9 by llpx_sn_lref, yle_fwd_plus_sn1/ + ] +| /2 width=1 by llpx_sn_free/ +| #L1 #L2 #lt #p #HL12 #X #l #m #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/ +| #a #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct + elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct + @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *) + @(IHT … HTU) /2 width=1 by yle_succ/ +| #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct + elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct + /3 width=4 by llpx_sn_flat/ +] +qed-. + +(**) (* the minor premise comes first *) +lemma llpx_sn_ge: ∀R,L1,L2,T,l1,l2. l1 ≤ l2 → + llpx_sn R l1 T L1 L2 → llpx_sn R l2 T L1 L2. +#R #L1 #L2 #T #l1 #l2 * -l1 -l2 (**) (* destructed yle *) +/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/ +qed-. + +lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 → + llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → + llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2. +/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-. + +lemma llpx_sn_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → + ∀L1,L2,T,l. llpx_sn R1 l T L1 L2 → llpx_sn R2 l T L1 L2. +#R1 #R2 #HR12 #L1 #L2 #T #l #H elim H -L1 -L2 -T -l +/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma new file mode 100644 index 000000000..f40046c5f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma @@ -0,0 +1,63 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/frees.ma". +include "basic_2/multiple/llpx_sn_alt_rec.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* alternative definition of llpx_sn (not recursive) *) +definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝ + λR,l,T,L1,L2. |L1| = |L2| ∧ + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ R K1 V1 V2 + ). + +(* Main properties **********************************************************) + +theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,l. llpx_sn R l T L1 L2 → llpx_sn_alt R l T L1 L2. +#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U +#x #IHx #L1 #U #Hx #L2 #l #H elim (llpx_sn_inv_alt_r … H) -H +#HL12 #IHU @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 elim (frees_inv … H) -H +[ -x #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/ +| * #J1 #K10 #W10 #j #Hlj #Hji #HnU #HLK10 yminus_inj >yminus_inj #HnW10 destruct + lapply (drop_fwd_drop2 … HLK10) #H + lapply (drop_conf_ge … H … HLK1 ?) -H /2 width=1 by ylt_fwd_le_succ1/ (minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus yminus_SO2 +#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *) +lapply (drop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1 +lapply (drop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2 +/4 width=9 by frees_be, yle_plus_dx2_trans, yle_succ_dx, ylt_inj/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt_rec.ma new file mode 100644 index 000000000..870a4bbef --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt_rec.ma @@ -0,0 +1,249 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/substitution/lift_neg.ma". +include "basic_2/substitution/drop_drop.ma". +include "basic_2/multiple/llpx_sn.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* alternative definition of llpx_sn (recursive) *) +inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝ +| llpx_sn_alt_r_intro: ∀L1,L2,T,l. + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2 + ) → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2 + ) → |L1| = |L2| → llpx_sn_alt_r R l T L1 L2 +. + +(* Compact definition of llpx_sn_alt_r **************************************) + +lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,l. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 + ) → llpx_sn_alt_r R l T L1 L2. +#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/ +qed. + +lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv. + (∀L1,L2,T,l. |L1| = |L2| → ( + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2 + ) → S l T L1 L2) → + ∀L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → S l T L1 L2. +#R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l +#L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/ +qed-. + +lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2. +#R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l +#L1 #L2 #T #l #HL12 #IH @conj // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2 → + llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R l T L1 L2. +#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H +#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 +elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 // +/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/ +qed-. + +lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 → + llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). +#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H +#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 +[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 + /3 width=9 by nlift_bind_sn, and3_intro/ +| lapply (yle_inv_succ1 … Hli) -Hli * #Hli #Hi (lift_inv_sort1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l + /2 width=1 by llpx_sn_sort/ +| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hli #H destruct + [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l + /2 width=1 by llpx_sn_skip/ + | elim (ylt_yle_false … Hil0) -L1 -L2 -K1 -K2 -m -Hil0 + /3 width=3 by yle_trans, yle_inj/ + ] +| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hli #H destruct [ -HK12 | -IHK12 ] + [ elim (drop_trans_lt … HLK1 … HK11) // -K1 + elim (drop_trans_lt … HLK2 … HK22) // -Hli -K2 + /3 width=18 by llpx_sn_lref/ + | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1 + lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hli -Hl0 -K2 + /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/ + ] +| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hil #H destruct + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12 + [ /3 width=7 by llpx_sn_free, drop_fwd_be/ + | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1 + lapply (drop_fwd_length … HLK2) -HLK2 #HLK2 + @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) + ] +| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l -m + /2 width=1 by llpx_sn_gref/ +| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H + #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/ +| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H + #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ +] +qed-. + +lemma llpx_sn_lift_ge: ∀R,K1,K2,T,l0. llpx_sn R l0 T K1 K2 → + ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀U. ⬆[l, m] T ≡ U → l ≤ l0 → llpx_sn R (l0+m) U L1 L2. +#R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0 +[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l + /2 width=1 by llpx_sn_sort/ +| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H + * #_ #H destruct + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 + [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/ + | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/ + ] +| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hil #H destruct + [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -m -Hil0 + /3 width=3 by ylt_yle_trans, ylt_inj/ + | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1 + lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hil -Hl0 -K2 + /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/ + ] +| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hil #H destruct + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12 + [ /3 width=7 by llpx_sn_free, drop_fwd_be/ + | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1 + lapply (drop_fwd_length … HLK2) -HLK2 #HLK2 + @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) + ] +| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l + /2 width=1 by llpx_sn_gref/ +| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H + #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_skip, yle_succ/ +| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H + #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/ +] +qed-. + +(* Inversion lemmas on relocation *******************************************) + +lemma llpx_sn_inv_lift_le: ∀R. d_deliftable_sn R → + ∀L1,L2,U,l0. llpx_sn R l0 U L1 L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 T K1 K2. +#R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 +[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m + /2 width=1 by llpx_sn_sort/ +| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H + * #_ #H destruct + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 + [ /2 width=1 by llpx_sn_skip/ + | /3 width=3 by llpx_sn_skip, yle_ylt_trans/ + ] +| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H + * #Hil #H destruct [ -HK12 | -IHK12 ] + [ elim (drop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1 + elim (drop_conf_lt … HLK2 … HLK22) // -Hil -L2 #L2 #V2 #HKL2 #HKL22 #HVW2 + elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12 + lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct + /3 width=10 by llpx_sn_lref/ + | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 + lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 + elim (yle_inv_plus_inj2 … Hil) -Hil /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *) + ] +| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H + * #_ #H destruct + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) + [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 + lapply (drop_fwd_length_le4 … HLK2) -HLK2 + #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) + | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H + lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H + /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ + ] +| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m + /2 width=1 by llpx_sn_gref/ +| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 … H) -H + #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/ +| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat2 … H) -H + #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ +] +qed-. + +lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l ≤ l0 → l0 ≤ l + m → llpx_sn R l T K1 K2. +#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 +[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m + /2 width=1 by llpx_sn_sort/ +| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H + * #Hil #H destruct + [ lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 + -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/ + | elim (ylt_yle_false … Hil0) -L1 -L2 -Hl0 -Hil0 + /3 width=3 by yle_trans, yle_inj/ (**) (* slow *) + ] +| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H + * #Hil #H destruct + [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0m -Hil0 + /3 width=3 by ylt_yle_trans, ylt_inj/ + | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 + lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0m + elim (yle_inv_plus_inj2 … Hil) -Hil /3 width=9 by llpx_sn_lref, yle_inj/ + ] +| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H + * #_ #H destruct + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) + [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 + lapply (drop_fwd_length_le4 … HLK2) -HLK2 + #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) + | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H + lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H + /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ + ] +| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m + /2 width=1 by llpx_sn_gref/ +| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_bind2 … H) -H + #V #T #HVW #HTU #H destruct + @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) + @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/ +| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_flat2 … H) -H + #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ +] +qed-. + +lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l + m ≤ l0 → llpx_sn R (l0-m) T K1 K2. +#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 +[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l + /2 width=1 by llpx_sn_sort/ +| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H + * #Hil #H destruct [ -Hil0 | -Hlml0 ] + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 + [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/ + | elim (yle_inv_plus_inj2 … Hil) -Hil + /3 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx/ + ] +| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H + * #Hil #H destruct + [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0 + /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/ + | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 + lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hlml0 -Hil + /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/ + ] +| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H + * #_ #H destruct + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) + [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 + lapply (drop_fwd_length_le4 … HLK2) -HLK2 + #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *) + | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H + lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H + /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ + ] +| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l + /2 width=1 by llpx_sn_gref/ +| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_bind2 … H) -H + #V #T #HVW #HTU #H destruct + @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) + yminus_Y_inj #K1 #HK12 #HLK1 + lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ +| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/ +| /4 width=1 by llpx_sn_bind, lreq_succ/ +] +qed-. + +lemma llpx_sn_lreq_trans: ∀R,L,L1,T,l. llpx_sn R l T L L1 → + ∀L2. L1 ⩬[l, ∞] L2 → llpx_sn R l T L L2. +#R #L #L1 #T #l #H elim H -L -L1 -T -l +/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/ +[ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12 + elim (lreq_drop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2 + lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ +| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/ +| /4 width=1 by llpx_sn_bind, lreq_succ/ +] +qed-. + +lemma llpx_sn_lreq_repl: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → ∀K1. K1 ⩬[l, ∞] L1 → + ∀K2. L2 ⩬[l, ∞] K2 → llpx_sn R l T K1 K2. +/3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-. + +lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) → + ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2). +#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT +/3 width=7 by llpx_sn_lreq_repl, lreq_succ/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_tc.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_tc.ma new file mode 100644 index 000000000..13fc14e82 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_tc.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2/multiple/llpx_sn_drop.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* Properties about transitive closure **************************************) + +lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) → + ∀I,L,V1,V2,T. LTC … R L V1 V2 → + LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2). +#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2 +/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/ +qed.