X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Flift.ma;h=b588b29b793e163cfed492afc3981b278e064fbd;hb=d833e40ce45e301a01ddd9ea66c29fb2b34bb685;hp=e8e23a5550f9c417cea2e631fe20f7ffca780677;hpb=35653f628dc3a3e665fee01acc19c660c9d555e3;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma index e8e23a555..b588b29b7 100644 --- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma @@ -13,8 +13,9 @@ (**************************************************************************) include "Basic_2/grammar/term_weight.ma". +include "Basic_2/grammar/term_simple.ma". -(* RELOCATION ***************************************************************) +(* BASIC TERM RELOCATION ****************************************************) (* Basic_1: includes: lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat @@ -36,14 +37,14 @@ interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2). (* Basic inversion lemmas ***************************************************) -fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2. +fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2. #d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/ qed. -lemma lift_inv_refl_O2: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2. +lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2. /2 width=4/ qed-. -fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k. +fact 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 @@ -51,10 +52,10 @@ fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k ] qed. -lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k. +lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k. /2 width=5/ qed-. -fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i → +fact 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 @@ -66,23 +67,23 @@ fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i → ] qed. -lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → +lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). /2 width=3/ qed-. -lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i. +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). +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-. -fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p. +fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p. #d #e #T1 #T2 * -d -e -T1 -T2 // [ #i #d #e #_ #k #H destruct | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct @@ -90,12 +91,12 @@ fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → ] qed. -lemma lift_inv_gref1: ∀d,e,T2,p. ↑[d,e] §p ≡ T2 → T2 = §p. +lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p. /2 width=5/ qed-. -fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → +fact 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 & + ∃∃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 @@ -107,14 +108,14 @@ fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ] 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 & +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 width=3/ qed-. -fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → +fact 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 & + ∃∃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 @@ -126,12 +127,12 @@ fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ] 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 & +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 width=3/ qed-. -fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k. +fact 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 @@ -140,10 +141,10 @@ fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k qed. (* Basic_1: was: lift_gen_sort *) -lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k. +lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k. /2 width=5/ qed-. -fact lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i → +fact 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 @@ -156,12 +157,12 @@ fact lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i → qed. (* Basic_1: was: lift_gen_lref *) -lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → +lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). /2 width=3/ qed-. (* Basic_1: was: lift_gen_lref_lt *) -lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i. +lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i. #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * // #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd elim (lt_inv_plus_l … Hdd) -Hdd #Hdd @@ -169,7 +170,7 @@ elim (lt_refl_false … Hdd) qed-. (* Basic_1: was: lift_gen_lref_false *) -lemma lift_inv_lref2_be: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → +lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d ≤ i → i < d + e → False. #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * [ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ] @@ -178,14 +179,14 @@ elim (lt_refl_false … H) qed-. (* Basic_1: was: lift_gen_lref_ge *) -lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e). +lemma lift_inv_lref2_ge: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e). #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * // #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd elim (lt_inv_plus_l … Hdd) -Hdd #Hdd elim (lt_refl_false … Hdd) qed-. -fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p. +fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p. #d #e #T1 #T2 * -d -e -T1 -T2 // [ #i #d #e #_ #k #H destruct | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct @@ -193,12 +194,12 @@ fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → ] qed. -lemma lift_inv_gref2: ∀d,e,T1,p. ↑[d,e] T1 ≡ §p → T1 = §p. +lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p. /2 width=5/ qed-. -fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → +fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀I,V2,U2. T2 = 𝕓{I} V2.U2 → - ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & + ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 & T1 = 𝕓{I} V1. U1. #d #e #T1 #T2 * -d -e -T1 -T2 [ #k #d #e #I #V2 #U2 #H destruct @@ -211,14 +212,14 @@ fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → qed. (* Basic_1: was: lift_gen_bind *) -lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 → - ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & +lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ 𝕓{I} V2. U2 → + ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 & T1 = 𝕓{I} V1. U1. /2 width=3/ qed-. -fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → +fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀I,V2,U2. T2 = 𝕗{I} V2.U2 → - ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & + ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 & T1 = 𝕗{I} V1. U1. #d #e #T1 #T2 * -d -e -T1 -T2 [ #k #d #e #I #V2 #U2 #H destruct @@ -231,12 +232,12 @@ fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → qed. (* Basic_1: was: lift_gen_flat *) -lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 → - ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & +lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ 𝕗{I} V2. U2 → + ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 & T1 = 𝕗{I} V1. U1. /2 width=3/ qed-. -lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ↑[d, e] 𝕔{I} V. T ≡ V → False. +lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] 𝕔{I} V. T ≡ V → False. #d #e #J #V elim V -V [ * #i #T #H [ lapply (lift_inv_sort2 … H) -H #H destruct @@ -250,7 +251,7 @@ lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ↑[d, e] 𝕔{I} V. T ≡ V → False. ] qed-. -lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ↑[d, e] 𝕔{I} V. T ≡ T → False. +lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] 𝕔{I} V. T ≡ T → False. #J #T elim T -T [ * #i #V #d #e #H [ lapply (lift_inv_sort2 … H) -H #H destruct @@ -266,29 +267,41 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #[T1] = #[T2]. +lemma tw_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → #[T1] = #[T2]. #d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize // qed-. +lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝕊[T1] → 𝕊[T2]. +#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // +#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H +elim (simple_inv_bind … H) +qed-. + +lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝕊[T2] → 𝕊[T1]. +#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // +#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H +elim (simple_inv_bind … H) +qed-. + (* Basic properties *********************************************************) (* Basic_1: was: lift_lref_gt *) -lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i. +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 ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/ qed. -lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ↑[d, e] #j ≡ #i. +lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i. /2 width=1/ qed-. (* Basic_1: was: lift_r *) -lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T. +lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T. #T elim T -T [ * #i // #d elim (lt_or_ge i d) /2 width=1/ | * /2 width=1/ ] qed. -lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2. +lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2. #T1 elim T1 -T1 [ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/ | * #I #V1 #T1 #IHV1 #IHT1 #d #e @@ -300,9 +313,9 @@ lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2. qed. (* Basic_1: was: lift_free (right to left) *) -lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → +lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 → ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 → - ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2. + ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2. #d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2 [ /3 width=3/ | #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_ @@ -321,7 +334,7 @@ lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → qed. (* Basic_1: was only: dnf_dec2 dnf_dec *) -lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ↑[d,e] T1 ≡ T2). +lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2). #T1 elim T1 -T1 [ * [1,3: /3 width=2/ ] #i #d #e elim (lt_dec i d) #Hid