X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Ftps.ma;h=8a5dc13d083afa67e25924771bdb769e7b881d04;hb=d833e40ce45e301a01ddd9ea66c29fb2b34bb685;hp=d46642d472e54c3fd497fd188b49fccf974ab617;hpb=35653f628dc3a3e665fee01acc19c660c9d555e3;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 index d46642d47..8a5dc13d0 100644 --- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma @@ -20,7 +20,7 @@ include "Basic_2/substitution/ldrop.ma". 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 + ⇩[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) @@ -34,8 +34,8 @@ interpretation "parallel substritution (term)" (* Basic properties *********************************************************) -lemma tps_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 → - ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ≫ T2. +lemma tps_lsubs_conf: ∀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 -L1 -T1 -T2 -d -e [ // | #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12 @@ -45,14 +45,14 @@ lemma tps_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 → ] qed. -lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T. +lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ▶ T. #T elim T -T // #I elim I -I /2 width=1/ qed. (* Basic_1: was: subst1_ex *) -lemma tps_full: ∀K,V,T1,L,d. ↓[0, d] L ≡ (K. 𝕓{Abbr} V) → - ∃∃T2,T. L ⊢ T1 [d, 1] ≫ T2 & ↑[d, 1] T ≡ T2. +lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. 𝕓{Abbr} V) → + ∃∃T2,T. L ⊢ T1 [d, 1] ▶ T2 & ⇧[d, 1] T ≡ T2. #K #V #T1 elim T1 -T1 [ * #i #L #d #HLK /2 width=4/ elim (lt_or_eq_or_gt i d) #Hid /3 width=4/ @@ -67,9 +67,9 @@ lemma tps_full: ∀K,V,T1,L,d. ↓[0, d] L ≡ (K. 𝕓{Abbr} V) → ] qed. -lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 → +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 [d2, e2] ▶ T2. #L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1 [ // | #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12 @@ -81,7 +81,7 @@ lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 → qed. lemma tps_weak_top: ∀L,T1,T2,d,e. - L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2. + L ⊢ T1 [d, e] ▶ T2 → L ⊢ T1 [d, |L| - d] ▶ T2. #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e [ // | #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW @@ -93,14 +93,14 @@ lemma tps_weak_top: ∀L,T1,T2,d,e. qed. lemma tps_weak_all: ∀L,T1,T2,d,e. - L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2. + 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. +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 -L -T1 -T2 -d -e [ /2 width=3/ | #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde @@ -124,11 +124,11 @@ qed. (* Basic inversion lemmas ***************************************************) -fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} → +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 & + ⇩[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 /2 width=1/ @@ -138,43 +138,43 @@ fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = ] qed. -lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 → +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 & + ⇩[O, i] L ≡ K. 𝕓{Abbr} V & + ⇧[O, i + 1] V ≡ T2 & I = LRef i. /2 width=3/ qed-. (* Basic_1: was: subst1_gen_sort *) -lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫ T2 → T2 = ⋆k. +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 → +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. + ⇩[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 width=1/ * #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/ qed-. -lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ≫ T2 → T2 = §p. +lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶ T2 → T2 = §p. #L #T2 #p #d #e #H elim (tps_inv_atom1 … H) -H // * #K #V #i #_ #_ #_ #_ #H destruct qed-. -fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → +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 & + ∃∃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 @@ -184,15 +184,15 @@ fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → ] 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 & +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 width=3/ qed-. -fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → +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 & + ∃∃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 @@ -202,12 +202,12 @@ fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → ] 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 & +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 width=3/ qed-. -fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2. +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 -L -T1 -T2 -d -e [ // | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct @@ -218,12 +218,12 @@ fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T ] qed. -lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫ T2 → T1 = T2. +lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ▶ T2 → T1 = T2. /2 width=6/ qed-. (* Basic forward lemmas *****************************************************) -lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → #[T1] ≤ #[T2]. +lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → #[T1] ≤ #[T2]. #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize /3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *) qed-.