(**************************************************************************) (* ___ *) (* ||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/notation/relations/psubst_6.ma". include "basic_2/grammar/genv.ma". include "basic_2/relocation/lsuby.ma". (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) (* activate genv *) inductive cpy: ynat → ynat → relation4 genv lenv term term ≝ | cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I}) | cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d+e → ⇩[i] L ≡ K.ⓑ{I}V → ⇧[0, i+1] V ≡ W → cpy d e G L (#i) W | cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e. cpy d e G L V1 V2 → cpy (⫯d) e G (L.ⓑ{I}V1) T1 T2 → cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) | cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e. cpy d e G L V1 V2 → cpy d e G L T1 T2 → cpy d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) . interpretation "context-sensitive extended ordinary substritution (term)" 'PSubst G L T1 d e T2 = (cpy d e G L T1 T2). (* Basic properties *********************************************************) lemma lsuby_cpy_trans: ∀G,d,e. lsub_trans … (cpy d e G) (lsuby d e). #G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e [ // | #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12 elim (lsuby_ldrop_trans_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/ | /4 width=1 by lsuby_succ, cpy_bind/ | /3 width=1 by cpy_flat/ ] qed-. lemma cpy_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶[d, e] T. #G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/ qed. (* Basic_1: was: subst1_ex *) lemma cpy_full: ∀I,G,K,V,T1,L,d. ⇩[d] L ≡ K.ⓑ{I}V → ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[d, 1] T2 & ⇧[d, 1] T ≡ T2. #I #G #K #V #T1 elim T1 -T1 [ * #i #L #d #HLK /2 width=4 by lift_sort, lift_gref, ex2_2_intro/ elim (lt_or_eq_or_gt i d) #Hid /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ destruct elim (lift_total V 0 (i+1)) #W #HVW elim (lift_split … HVW i i) /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/ | * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 [ elim (IHU1 (L.ⓑ{J}W1) (d+1)) -IHU1 /3 width=9 by cpy_bind, ldrop_drop, lift_bind, ex2_2_intro/ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/ ] ] qed-. lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T2 → ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T2. #G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 // [ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/ | /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/ | /3 width=1 by cpy_flat/ ] qed-. lemma cpy_weak_top: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶[d, |L| - d] T2. #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e // [ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW lapply (ldrop_fwd_length_lt2 … HLK) /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/ | #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *) /2 width=1 by cpy_bind/ | /2 width=1 by cpy_flat/ ] qed-. lemma cpy_weak_full: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2. #G #L #T1 #T2 #d #e #HT12 lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12 /2 width=2 by cpy_weak_top/ qed-. lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀i. i ≤ d + e → ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d, i-d] T & ⦃G, L⦄ ⊢ T ▶[i, d+e-i] T2. #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e [ /2 width=3 by ex2_intro/ | #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde elim (ylt_split i j) [ -Hide -Hjde | -Hdi ] /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/ | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide elim (IHV12 i) -IHV12 // #V elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide >yplus_SO2 >yplus_succ1 #T #HT1 #HT2 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/ | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide /3 width=5 by ex2_intro, cpy_flat/ ] qed-. lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀i. i ≤ d + e → ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶[d, i-d] T2. #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e [ /2 width=3 by ex2_intro/ | #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde elim (ylt_split i j) [ -Hide -Hjde | -Hdi ] /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/ | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide elim (IHV12 i) -IHV12 // #V elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide >yplus_SO2 >yplus_succ1 #T #HT1 #HT2 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/ | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide /3 width=5 by ex2_intro, cpy_flat/ ] qed-. (* Basic forward lemmas *****************************************************) lemma cpy_fwd_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 → d ≤ dt → d + e ≤ dt + et → ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2. #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et [ * #i #G #L #dt #et #T1 #d #e #H #_ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ ] | #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ] [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/ | elim (le_inv_plus_l … Hid) #Hdie #Hei elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie #T2 #_ >plus_minus // ymax_pre_sn_comm // (**) (* explicit constructor *) ] | #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/ yplus_SO2 >yplus_succ1 >yplus_succ1 /3 width=2 by cpy_bind, lift_bind, ex2_intro/ | #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12 /3 width=2 by cpy_flat, lift_flat, ex2_intro/ ] qed-. lemma cpy_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ♯{T1} ≤ ♯{T2}. #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize /3 width=1 by monotonic_le_plus_l, le_plus/ qed-. (* Basic inversion lemmas ***************************************************) fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀J. T1 = ⓪{J} → T2 = ⓪{J} ∨ ∃∃I,K,V,i. d ≤ yinj i & i < d + e & ⇩[i] L ≡ K.ⓑ{I}V & ⇧[O, i+1] V ≡ T2 & J = LRef i. #G #L #T1 #T2 #d #e * -G -L -T1 -T2 -d -e [ #I #G #L #d #e #J #H destruct /2 width=1 by or_introl/ | #I #G #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/ | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct ] qed-. lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶[d, e] T2 → T2 = ⓪{I} ∨ ∃∃J,K,V,i. d ≤ yinj i & i < d + e & ⇩[i] L ≡ K.ⓑ{J}V & ⇧[O, i+1] V ≡ T2 & I = LRef i. /2 width=4 by cpy_inv_atom1_aux/ qed-. (* Basic_1: was: subst1_gen_sort *) lemma cpy_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶[d, e] T2 → T2 = ⋆k. #G #L #T2 #k #d #e #H elim (cpy_inv_atom1 … H) -H // * #I #K #V #i #_ #_ #_ #_ #H destruct qed-. (* Basic_1: was: subst1_gen_lref *) lemma cpy_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶[d, e] T2 → T2 = #i ∨ ∃∃I,K,V. d ≤ i & i < d + e & ⇩[i] L ≡ K.ⓑ{I}V & ⇧[O, i+1] V ≡ T2. #G #L #T2 #i #d #e #H elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/ * #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/ qed-. lemma cpy_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶[d, e] T2 → T2 = §p. #G #L #T2 #p #d #e #H elim (cpy_inv_atom1 … H) -H // * #I #K #V #i #_ #_ #_ #_ #H destruct qed-. fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 → ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 & ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯d, e] T2 & U2 = ⓑ{a,I}V2.T2. #G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e [ #I #G #L #d #e #b #J #W1 #U1 #H destruct | #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #b #J #W1 #U1 #H destruct | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #b #J #W1 #U1 #H destruct ] qed-. lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[d, e] U2 → ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯d, e] T2 & U2 = ⓑ{a,I}V2.T2. /2 width=3 by cpy_inv_bind1_aux/ qed-. fact cpy_inv_flat1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 → ∀I,V1,T1. U1 = ⓕ{I}V1.T1 → ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 & U2 = ⓕ{I}V2.T2. #G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e [ #I #G #L #d #e #J #W1 #U1 #H destruct | #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #J #W1 #U1 #H destruct | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #W1 #U1 #H destruct | #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ ] qed-. lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[d, e] U2 → ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 & U2 = ⓕ{I}V2.T2. /2 width=3 by cpy_inv_flat1_aux/ qed-. fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → e = 0 → T1 = T2. #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e [ // | #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct elim (ylt_yle_false … Hdi) -Hdi // | /3 width=1 by eq_f2/ | /3 width=1 by eq_f2/ ] qed-. lemma cpy_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶[d, 0] T2 → T1 = T2. /2 width=6 by cpy_inv_refl_O2_aux/ qed-. (* Basic_1: was: subst1_gen_lift_eq *) lemma cpy_inv_lift1_eq: ∀G,T1,U1,d,e. ⇧[d, e] T1 ≡ U1 → ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 → U1 = U2. #G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1 /2 width=4 by cpy_inv_refl_O2/ qed-. (* Basic_1: removed theorems 25: subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt subst0_confluence_neq subst0_confluence_eq subst0_tlt_head subst0_confluence_lift subst0_tlt subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift *)