X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Ftps.ma;fp=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Ftps.ma;h=7197a53d99dc1767c80bdf85d8ad19d95027e95b;hb=55dc00c1c44cc21c7ae179cb9df03e7446002c46;hp=0000000000000000000000000000000000000000;hpb=5b03651298a3943b67f49fb78dc30bb8b2780f30;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma new file mode 100644 index 000000000..7197a53d9 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma @@ -0,0 +1,221 @@ +(**************************************************************************) +(* ___ *) +(* ||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-2/grammar/cl_weight.ma". +include "Basic-2/substitution/drop.ma". + +(* PARALLEL SUBSTITUTION ON TERMS *******************************************) + +inductive tps: nat → nat → lenv → relation term ≝ +| tps_atom : ∀L,I,d,e. tps d e L (𝕒{I}) (𝕒{I}) +| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e → + ↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps d e L (#i) W +| tps_bind : ∀L,I,V1,V2,T1,T2,d,e. + tps d e L V1 V2 → tps (d + 1) e (L. 𝕓{I} V2) T1 T2 → + tps d e L (𝕓{I} V1. T1) (𝕓{I} V2. T2) +| tps_flat : ∀L,I,V1,V2,T1,T2,d,e. + tps d e L V1 V2 → tps d e L T1 T2 → + tps d e L (𝕗{I} V1. T1) (𝕗{I} V2. T2) +. + +interpretation "parallel substritution (term)" + 'PSubst L T1 d e T2 = (tps d e L T1 T2). + +(* Basic properties *********************************************************) + +lemma tps_leq_repl_dx: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 → + ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2. +#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e +[ // +| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12 + elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/ +| /4/ +| /3/ +] +qed. + +lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T. +#T elim T -T // +#I elim I -I /2/ +qed. + +lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → + L ⊢ T1 [d2, e2] ≫ T2. +#L #T1 #T2 #d1 #e1 #H elim H -H L T1 T2 d1 e1 +[ // +| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12 + lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2 + lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2/ +| /4/ +| /4/ +] +qed. + +lemma tps_weak_top: ∀L,T1,T2,d,e. + L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2. +#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e +[ // +| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW + lapply (drop_fwd_drop2_length … HLK) #Hi + lapply (le_to_lt_to_lt … Hdi Hi) #Hd + lapply (plus_minus_m_m_comm (|L|) d ?) /2/ +| normalize /2/ +| /2/ +] +qed. + +lemma tps_weak_all: ∀L,T1,T2,d,e. + L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2. +#L #T1 #T2 #d #e #HT12 +lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tps_weak_top … HT12) // +qed. + +lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e → + ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2. +#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e +[ /2/ +| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde + elim (lt_or_ge i j) + [ -Hide Hjde; + >(plus_minus_m_m_comm j d) in ⊢ (% → ?) // -Hdj /3/ + | -Hdi Hdj; #Hid + generalize in match Hide -Hide (**) (* rewriting in the premises, rewrites in the goal too *) + >(plus_minus_m_m_comm … Hjde) in ⊢ (% → ?) -Hjde /4/ + ] +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide + elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2 + elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *) + -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2 + lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1 /3 width=5/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide + elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 // + -Hdi Hide /3 width=5/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} → + T2 = 𝕒{I} ∨ + ∃∃K,V,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V & + ↑[O, i + 1] V ≡ T2 & + I = LRef i. +#L #T1 #T2 #d #e * -L T1 T2 d e +[ #L #I #d #e #J #H destruct -I /2/ +| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct -I /3 width=8/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct +] +qed. + +lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 → + T2 = 𝕒{I} ∨ + ∃∃K,V,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V & + ↑[O, i + 1] V ≡ T2 & + I = LRef i. +/2/ qed. + + +(* Basic-1: was: subst1_gen_sort *) +lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫ T2 → T2 = ⋆k. +#L #T2 #k #d #e #H +elim (tps_inv_atom1 … H) -H // +* #K #V #i #_ #_ #_ #_ #H destruct +qed. + +(* Basic-1: was: subst1_gen_lref *) +lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 → + T2 = #i ∨ + ∃∃K,V. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V & + ↑[O, i + 1] V ≡ T2. +#L #T2 #i #d #e #H +elim (tps_inv_atom1 … H) -H /2/ +* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct -i /3/ +qed. + +fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → + ∀I,V1,T1. U1 = 𝕓{I} V1. T1 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & + L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 & + U2 = 𝕓{I} V2. T2. +#d #e #L #U1 #U2 * -d e L U1 U2 +[ #L #k #d #e #I #V1 #T1 #H destruct +| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/ +| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct +] +qed. + +lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & + L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 & + U2 = 𝕓{I} V2. T2. +/2/ qed. + +fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → + ∀I,V1,T1. U1 = 𝕗{I} V1. T1 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 & + U2 = 𝕗{I} V2. T2. +#d #e #L #U1 #U2 * -d e L U1 U2 +[ #L #k #d #e #I #V1 #T1 #H destruct +| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/ +] +qed. + +lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 & + U2 = 𝕗{I} V2. T2. +/2/ qed. + +fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2. +#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e +[ // +| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct -e; + lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi Hide