X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fsubstitution%2Fcpy.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fsubstitution%2Fcpy.ma;h=0000000000000000000000000000000000000000;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=4961f85412ea0a8dfcec9e3233fc9730faf84156;hpb=277fc8ff21ce3dbd6893b1994c55cf5c06a98355;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy.ma deleted file mode 100644 index 4961f8541..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy.ma +++ /dev/null @@ -1,296 +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_2A/ynat/ynat_max.ma". -include "basic_2A/notation/relations/psubst_6.ma". -include "basic_2A/grammar/genv.ma". -include "basic_2A/substitution/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,l,m. cpy l m G L (⓪{I}) (⓪{I}) -| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m → - ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W -| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. - cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 → - cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) -| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m. - cpy l m G L V1 V2 → cpy l m G L T1 T2 → - cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) -. - -interpretation "context-sensitive extended ordinary substritution (term)" - 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2). - -(* Basic properties *********************************************************) - -lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy 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 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12 - elim (lsuby_drop_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,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] 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,l. ⬇[l] L ≡ K.ⓑ{I}V → - ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2. -#I #G #K #V #T1 elim T1 -T1 -[ * #i #L #l #HLK - /2 width=4 by lift_sort, lift_gref, ex2_2_intro/ - elim (lt_or_eq_or_gt i l) #Hil - /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 #l #HLK - elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1 - /3 width=9 by cpy_bind, drop_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,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 elim H -G -L -T1 -T2 -l1 -m1 // -[ /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,l,m. - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m // -[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW - lapply (drop_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,l,m. - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2. -#G #L #T1 #T2 #l #m #HT12 -lapply (cpy_weak … HT12 0 (l + m) ? ?) -HT12 -/2 width=2 by cpy_weak_top/ -qed-. - -lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀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 elim H -G -L -T1 -T2 -l -m -[ /2 width=3 by ex2_intro/ -| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm - elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ] - /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/ -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm - elim (IHV12 i) -IHV12 // #V - elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm - >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 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm - elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm - /3 width=5 by ex2_intro, cpy_flat/ -] -qed-. - -lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -[ /2 width=3 by ex2_intro/ -| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm - elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ] - /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/ -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm - elim (IHV12 i) -IHV12 // #V - elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm - >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 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm - elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm - /3 width=5 by ex2_intro, cpy_flat/ -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cpy_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 elim H -G -L -U1 -U2 -lt -mt -[ * #i #G #L #lt #mt #T1 #l #m #H #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 … H) -H * #Hil #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 #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt - elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ] - [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/ - | elim (le_inv_plus_l … Hil) #Hlim #Hmi - elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim - #T2 #_ >plus_minus // ymax_pre_sn_comm // (**) (* explicit constructor *) - ] -| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt - 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 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt - 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,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize -/3 width=1 by monotonic_le_plus_l, le_plus/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} → - T2 = ⓪{J} ∨ - ∃∃I,K,V,i. l ≤ yinj i & i < l + m & - ⬇[i] L ≡ K.ⓑ{I}V & - ⬆[O, i+1] V ≡ T2 & - J = LRef i. -#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m -[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/ -| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/ -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct -| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct -] -qed-. - -lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 → - T2 = ⓪{I} ∨ - ∃∃J,K,V,i. l ≤ yinj i & i < l + m & - ⬇[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,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k. -#G #L #T2 #k #l #m #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,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 → - T2 = #i ∨ - ∃∃I,K,V. l ≤ i & i < l + m & - ⬇[i] L ≡ K.ⓑ{I}V & - ⬆[O, i+1] V ≡ T2. -#G #L #T2 #i #l #m #H -elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/ -* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/ -qed-. - -lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p. -#G #L #T2 #p #l #m #H -elim (cpy_inv_atom1 … H) -H // -* #I #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → - ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & - ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 & - U2 = ⓑ{a,I}V2.T2. -#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m -[ #I #G #L #l #m #b #J #W1 #U1 #H destruct -| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ -| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct -] -qed-. - -lemma cpy_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. -/2 width=3 by cpy_inv_bind1_aux/ qed-. - -fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → - ∀I,V1,T1. U1 = ⓕ{I}V1.T1 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 & - U2 = ⓕ{I}V2.T2. -#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m -[ #I #G #L #l #m #J #W1 #U1 #H destruct -| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct -| #I #G #L #V1 #V2 #T1 #T2 #l #m #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,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. -/2 width=3 by cpy_inv_flat1_aux/ qed-. - - -fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -[ // -| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct - elim (ylt_yle_false … Hli) -Hli // -| /3 width=1 by eq_f2/ -| /3 width=1 by eq_f2/ -] -qed-. - -lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 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,l,m. ⬆[l, m] T1 ≡ U1 → - ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2. -#G #T1 #U1 #l #m #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 -*)