X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fmultiple%2Fcpys_lift.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2A%2Fmultiple%2Fcpys_lift.ma;h=d3f292a7fba03fdc24fe995fbd876136822ef27c;hb=d2545ffd201b1aa49887313791386add78fa8603;hp=0000000000000000000000000000000000000000;hpb=57ae1762497a5f3ea75740e2908e04adb8642cc2;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma new file mode 100644 index 000000000..d3f292a7f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma @@ -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_2A/substitution/cpy_lift.ma". +include "basic_2A/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_succ/ +] +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_succ, 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 // +| * #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 ≤ yinj 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 ≤ yinj 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. yinj 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 → yinj 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 → + yinj 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 ≤ yinj l + m → yinj l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj 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 ≤ yinj 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 ≤ yinj 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-.