X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Flift_lift.ma;h=30bf8886e2485351bba39599feb160468e373219;hb=d833e40ce45e301a01ddd9ea66c29fb2b34bb685;hp=f9c99457db747c0e0c0b1983e4873c2aa52235fe;hpb=55dc00c1c44cc21c7ae179cb9df03e7446002c46;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift_lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift_lift.ma index f9c99457d..30bf8886e 100644 --- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift_lift.ma +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift_lift.ma @@ -12,148 +12,189 @@ (* *) (**************************************************************************) -include "Basic-2/substitution/lift.ma". +include "Basic_2/substitution/lift.ma". -(* RELOCATION ***************************************************************) +(* BASIC TERM RELOCATION ****************************************************) (* Main properies ***********************************************************) -(* Basic-1: was: lift_inj *) -theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2. -#d #e #T1 #U #H elim H -H d e T1 U +(* Basic_1: was: lift_inj *) +theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2. +#d #e #T1 #U #H elim H -d -e -T1 -U [ #k #d #e #X #HX lapply (lift_inv_sort2 … HX) -HX // -| #i #d #e #Hid #X #HX +| #i #d #e #Hid #X #HX lapply (lift_inv_lref2_lt … HX ?) -HX // -| #i #d #e #Hdi #X #HX - lapply (lift_inv_lref2_ge … HX ?) -HX /2/ +| #i #d #e #Hdi #X #HX + lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/ +| #p #d #e #X #HX + lapply (lift_inv_gref2 … HX) -HX // | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/ + elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/ + elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ ] -qed. +qed-. -(* Basic-1: was: lift_gen_lift *) -theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T → +(* Basic_1: was: lift_gen_lift *) +theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T → d1 ≤ d2 → - ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1. -#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T + ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 - lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/ + lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/ | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 - lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/ + lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/ | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12 - elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2 - [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/ - | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2 - @(ex2_1_intro … #(i - e2)) - [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ] - | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/ - ] + elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct + [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/ + | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H + elim (le_inv_plus_l … H) -H #Hide2 #He2i + lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12 + >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i + /4 width=3/ ] +| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 + lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/ | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 - lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2; - elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1 - >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/ + lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct + elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1 + >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /3 width=5/ | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 - lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2; - elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1 - elim (IHU … HU2 ?) /3 width = 5/ + lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct + elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1 + elim (IHU … HU2 ?) // /3 width=5/ +] +qed. + +(* Note: apparently this was missing in Basic_1 *) +theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T → + e ≤ e1 → e1 ≤ e + e2 → + ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T +[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/ +| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2 + >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ] +| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2 + elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2 + [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/ + | >(lift_inv_lref2_ge … H ?) -H // + lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i + elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1 + @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12 + @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ] + ] +| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/ +| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2 + elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2 + elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/ +| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2 + elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + elim (IHV1 … HV2 ? ?) -V // + elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/ ] qed. -theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2. -#d #e #T #U1 #H elim H -H d e T U1 +theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2. +#d #e #T #U1 #H elim H -d -e -T -U1 [ #k #d #e #X #HX lapply (lift_inv_sort1 … HX) -HX // | #i #d #e #Hid #X #HX lapply (lift_inv_lref1_lt … HX ?) -HX // | #i #d #e #Hdi #X #HX lapply (lift_inv_lref1_ge … HX ?) -HX // +| #p #d #e #X #HX + lapply (lift_inv_gref1 … HX) -HX // | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/ + elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX - elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/ + elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/ ] -qed. +qed-. -(* Basic-1: was: lift_free (left to right) *) -theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → - d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2. -#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T +(* Basic_1: was: lift_free (left to right) *) +theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → + d1 ≤ d2 → d2 ≤ d1 + e1 → ⇧[d1, e1 + e2] T1 ≡ T2. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ >(lift_inv_sort1 … HT2) -HT2 // | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 - lapply (lift_inv_lref1_lt … HT2 Hid2) /2/ + lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1/ | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21 lapply (lift_inv_lref1_ge … HT2 ?) -HT2 - [ @(transitive_le … Hd21 ?) -Hd21 /2/ - | -Hd21 /2/ + [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/ + | -Hd21 /2 width=1/ ] +| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ + >(lift_inv_gref1 … HT2) -HT2 // | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 - elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; - lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10 - lapply (IHT12 … HT20 ? ?) /2/ + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10 + lapply (IHT12 … HT20 ? ?) /2 width=1/ | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 - elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; - lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10 - lapply (IHT12 … HT20 ? ?) /2/ + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10 + lapply (IHT12 … HT20 ? ?) // /2 width=1/ ] qed. -(* Basic-1: was: lift_d (right to left) *) -theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 → - ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2. -#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T +(* Basic_1: was: lift_d (right to left) *) +theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d2 ≤ d1 → + ∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T0 ≡ T2. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T [ #k #d1 #e1 #d2 #e2 #X #HX #_ - >(lift_inv_sort1 … HX) -HX /2/ + >(lift_inv_sort1 … HX) -HX /2 width=3/ | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2 - elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/ + elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /3 width=3/ /4 width=3/ | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2 - lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X; - >plus_plus_comm_23 /4/ + lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct + >plus_plus_comm_23 /4 width=3/ +| #p #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_gref1 … HX) -HX /2 width=3/ | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 - elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; - elim (IHV12 … HV20 ?) -IHV12 HV20 // - elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/ + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + elim (IHV12 … HV20 ?) -IHV12 -HV20 // + elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /3 width=5/ | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 - elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; - elim (IHV12 … HV20 ?) -IHV12 HV20 // - elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/ + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct + elim (IHV12 … HV20 ?) -IHV12 -HV20 // + elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/ ] qed. -(* Basic-1: was: lift_d (left to right) *) -theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → - ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2. -#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T +(* Basic_1: was: lift_d (left to right) *) +theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → + ∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[d1, e1] T0 ≡ T2. +#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T [ #k #d1 #e1 #d2 #e2 #X #HX #_ - >(lift_inv_sort1 … HX) -HX /2/ + >(lift_inv_sort1 … HX) -HX /2 width=3/ | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e - lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e - lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2 - lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/ + lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1/ #Hid2e + lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2 + lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/ | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ - elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X; - [2: >plus_plus_comm_23] /4/ + elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/ +| #p #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_gref1 … HX) -HX /2 width=3/ | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded - elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; - elim (IHV12 … HV20 ?) -IHV12 HV20 // - elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T -