X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Fcpys_lift.ma;h=4a621d7d6653a82753fd6693456c460fca06bd0b;hb=e5378812c068074f0ac627541d3f066e135c8824;hp=c4b9518e6c6f1595e02256a52fd228fe4ec42199;hpb=928cfe1ebf2fbd31731c8851cdec70802596016d;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma index c4b9518e6..4a621d7d6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma @@ -21,14 +21,14 @@ include "basic_2/substitution/cpys.ma". lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e. d ≤ yinj i → i < d + e → - ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ⫰(d+e-i)] U1 → + ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ⫰(d+e-i)] U1 → ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2. #I #G #L #K #V #U1 #i #d #e #Hdi #Hide #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 (ldrop_fwd_ldrop2 … HLK) -HLK #HLK + lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 lapply (cpy_weak … HU02 d e ? ?) -HU02 [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] @@ -38,7 +38,7 @@ qed. lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,d. d ≤ yinj i → - ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ∞] U1 → + ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ∞] U1 → ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, ∞] U2. #I #G #L #K #V #U1 #i #d #Hdi #HLK #HVU1 #U2 #HU12 @(cpys_subst … HLK … HU12) >yminus_Y_inj // @@ -49,7 +49,7 @@ qed. lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 → T2 = ⓪{I} ∨ ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e & - ⇩[O, i] L ≡ K.ⓑ{J}V1 & + ⇩[i] L ≡ K.ⓑ{J}V1 & ⦃G, K⦄ ⊢ V1 ▶*×[0, ⫰(d+e-i)] V2 & ⇧[O, i+1] V2 ≡ T2 & I = LRef i. @@ -59,7 +59,7 @@ lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 → [ #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 #Hdi #Hide #HLK #HV1 #HVT #HI - lapply (ldrop_fwd_ldrop2 … HLK) #H + lapply (ldrop_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/ @@ -70,7 +70,7 @@ qed-. lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 → T2 = #i ∨ ∃∃I,K,V1,V2. d ≤ i & i < d + e & - ⇩[O, i] L ≡ K.ⓑ{I}V1 & + ⇩[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ▶*×[0, ⫰(d+e-i)] V2 & ⇧[O, i+1] V2 ≡ T2. #G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ @@ -78,7 +78,7 @@ lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 → qed-. lemma cpys_inv_lref1_ldrop: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 → - ∀I,K,V1. ⇩[O, i] L ≡ K.ⓑ{I}V1 → + ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 → ∀V2. ⇧[O, i+1] V2 ≡ T2 → ∧∧ ⦃G, K⦄ ⊢ V1 ▶*×[0, ⫰(d+e-i)] V2 & d ≤ i @@ -95,10 +95,10 @@ qed-. (* Properties on relocation *************************************************) lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 → - ∀L,U1,d,e. dt + et ≤ yinj d → ⇩[d, e] L ≡ K → + ∀L,U1,s,d,e. dt + et ≤ yinj d → ⇩[s, d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2. -#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2 +#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2 [ #U2 #H >(lift_mono … HTU1 … H) -H // | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 elim (lift_total T d e) #U #HTU @@ -108,10 +108,10 @@ lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 → qed-. lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 → - ∀L,U1,d,e. dt ≤ yinj d → d ≤ dt + et → - ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → + ∀L,U1,s,d,e. dt ≤ yinj d → d ≤ dt + et → + ⇩[s, d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et + e] U2. -#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2 +#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2 [ #U2 #H >(lift_mono … HTU1 … H) -H // | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 elim (lift_total T d e) #U #HTU @@ -121,10 +121,10 @@ lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 → qed-. lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 → - ∀L,U1,d,e. yinj d ≤ dt → ⇩[d, e] L ≡ K → + ∀L,U1,s,d,e. yinj d ≤ dt → ⇩[s, d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt+e, et] U2. -#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2 +#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2 [ #U2 #H >(lift_mono … HTU1 … H) -H // | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 elim (lift_total T d e) #U #HTU @@ -136,10 +136,10 @@ qed-. (* Inversion lemmas for relocation ******************************************) lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → dt + et ≤ d → ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 & ⇧[d, e] T2 ≡ U2. -#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind … H) -U2 +#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdetd @(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/ @@ -147,10 +147,10 @@ lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 qed-. lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt ≤ d → d + e ≤ dt + et → + ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt ≤ d → yinj d + e ≤ dt + et → ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2. -#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind … H) -U2 +#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(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/ @@ -158,10 +158,10 @@ lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 qed-. lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d + e ≤ dt → + ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + yinj d + e ≤ dt → ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2. -#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2 +#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdedt @(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/ @@ -171,11 +171,11 @@ qed-. (* Advanced inversion lemmas on relocation **********************************) lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[d, dt + et - (d + e)] T2 & + ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[d, dt + et - (yinj d + e)] T2 & ⇧[d, e] T2 ≡ U2. -#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind … H) -U2 +#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(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/ @@ -183,10 +183,10 @@ lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] qed-. lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt ≤ d → dt + et ≤ d + e → + ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt ≤ d → dt + et ≤ yinj d + e → ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2. -#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind … H) -U2 +#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(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/ @@ -194,10 +194,10 @@ lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] qed-. lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 → - ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → - dt ≤ d → d ≤ dt + et → dt + et ≤ d + e → + ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → + dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e → ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2. -#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind … H) -U2 +#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(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/