X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Flib%2Flambda-delta%2Fsubstitution%2Flift_defs.ma;fp=matita%2Fmatita%2Flib%2Flambda-delta%2Fsubstitution%2Flift_defs.ma;h=0000000000000000000000000000000000000000;hb=baccd5a2f3b79c295b1f9444575bfb351577634e;hp=71d438d6899debfa4b5e26f68c416b24ca1ac293;hpb=1cd2f9aa6e0aee9eb4939b39c985b6ad6605092b;p=helm.git diff --git a/matita/matita/lib/lambda-delta/substitution/lift_defs.ma b/matita/matita/lib/lambda-delta/substitution/lift_defs.ma deleted file mode 100644 index 71d438d68..000000000 --- a/matita/matita/lib/lambda-delta/substitution/lift_defs.ma +++ /dev/null @@ -1,211 +0,0 @@ -(* - ||M|| This file is part of HELM, an Hypertextual, Electronic - ||A|| Library of Mathematics, developed at the Computer Science - ||T|| Department of the University of Bologna, Italy. - ||I|| - ||T|| - ||A|| This file is distributed under the terms of the - \ / GNU General Public License Version 2 - \ / - V_______________________________________________________________ *) - -include "lambda-delta/syntax/term.ma". - -(* RELOCATION ***************************************************************) - -inductive lift: term → nat → nat → term → Prop ≝ -| lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k) -| lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i) -| lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e)) -| lift_bind : ∀I,V1,V2,T1,T2,d,e. - lift V1 d e V2 → lift T1 (d + 1) e T2 → - lift (𝕓{I} V1. T1) d e (𝕓{I} V2. T2) -| lift_flat : ∀I,V1,V2,T1,T2,d,e. - lift V1 d e V2 → lift T1 d e T2 → - lift (𝕗{I} V1. T1) d e (𝕗{I} V2. T2) -. - -interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2). - -(* Basic properties *********************************************************) - -lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i. -#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/ -qed. - -lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T. -#T elim T -T -[ // -| #i #d elim (lt_or_ge i d) /2/ -| #I elim I -I /2/ -] -qed. - -lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2. -#T1 elim T1 -T1 -[ /2/ -| #i #d #e elim (lt_or_ge i d) /3/ -| * #I #V1 #T1 #IHV1 #IHT1 #d #e - elim (IHV1 d e) -IHV1 #V2 #HV12 - [ elim (IHT1 (d+1) e) -IHT1 /3/ - | elim (IHT1 d e) -IHT1 /3/ - ] -] -qed. - -(* Basic inversion lemmas ***************************************************) - -lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2. -#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/ -qed. - -lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2. -/2/ qed. - -lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k. -#d #e #T1 #T2 * -d e T1 T2 // -[ #i #d #e #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -] -qed. - -lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k. -/2 width=5/ qed. - -lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i → - (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). -#d #e #T1 #T2 * -d e T1 T2 -[ #k #d #e #i #H destruct -| #j #d #e #Hj #i #Hi destruct /3/ -| #j #d #e #Hj #i #Hi destruct /3/ -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct -] -qed. - -lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → - (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). -/2/ qed. - -lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i. -#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // -#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd -elim (lt_refl_false … Hdd) -qed. - -lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e). -#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // -#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd -elim (lt_refl_false … Hdd) -qed. - -lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → - ∀I,V1,U1. T1 = 𝕓{I} V1.U1 → - ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & - T2 = 𝕓{I} V2. U2. -#d #e #T1 #T2 * -d e T1 T2 -[ #k #d #e #I #V1 #U1 #H destruct -| #i #d #e #_ #I #V1 #U1 #H destruct -| #i #d #e #_ #I #V1 #U1 #H destruct -| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/ -| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct -] -qed. - -lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 → - ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & - T2 = 𝕓{I} V2. U2. -/2/ qed. - -lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → - ∀I,V1,U1. T1 = 𝕗{I} V1.U1 → - ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & - T2 = 𝕗{I} V2. U2. -#d #e #T1 #T2 * -d e T1 T2 -[ #k #d #e #I #V1 #U1 #H destruct -| #i #d #e #_ #I #V1 #U1 #H destruct -| #i #d #e #_ #I #V1 #U1 #H destruct -| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct -| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/ -] -qed. - -lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 → - ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & - T2 = 𝕗{I} V2. U2. -/2/ qed. - -lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k. -#d #e #T1 #T2 * -d e T1 T2 // -[ #i #d #e #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct -] -qed. - -lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k. -/2 width=5/ qed. - -lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i → - (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). -#d #e #T1 #T2 * -d e T1 T2 -[ #k #d #e #i #H destruct -| #j #d #e #Hj #i #Hi destruct /3/ -| #j #d #e #Hj #i #Hi destruct