X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fmultiple%2Fllpx_sn_drop.ma;h=de36874c96b6794a5aaf8199174eb0d84778d12f;hb=c60524dec7ace912c416a90d6b926bee8553250b;hp=cc21b8b2ef55d77096938061bf1ec385d8013c7d;hpb=f10cfe417b6b8ec1c7ac85c6ecf5fb1b3fdf37db;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma index cc21b8b2e..de36874c9 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma @@ -13,39 +13,39 @@ (**************************************************************************) include "basic_2/substitution/drop_drop.ma". -include "basic_2/multiple/llpx_sn_leq.ma". +include "basic_2/multiple/llpx_sn_lreq.ma". (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) (* Advanced forward lemmas **************************************************) -lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → +lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → ∀I,K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 → - i < d ∨ + i < l ∨ ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 & - R K1 V1 V2 & d ≤ i. -#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ] + R K1 V1 V2 & l ≤ i. +#R #L1 #L2 #l #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ] [ #_ #H elim (lt_refl_false i) lapply (drop_fwd_length_lt2 … HLK2) -HLK2 /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *) | /2 width=1 by or_introl/ -| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi +| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli lapply (drop_mono … HLK22 … HLK2) -L2 #H destruct /3 width=5 by ex4_2_intro, or_intror/ ] qed-. -lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → +lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → ∀I,K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 → - i < d ∨ + i < l ∨ ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 & - R K1 V1 V2 & d ≤ i. -#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ] + R K1 V1 V2 & l ≤ i. +#R #L1 #L2 #l #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ] [ #H #_ elim (lt_refl_false i) lapply (drop_fwd_length_lt2 … HLK1) -HLK1 /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *) | /2 width=1 by or_introl/ -| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi +| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli lapply (drop_mono … HLK11 … HLK1) -L1 #H destruct /3 width=5 by ex4_2_intro, or_intror/ ] @@ -53,55 +53,55 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i → +lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i → ∀I,K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 → ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2. -#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 -[ #H elim (ylt_yle_false … H Hdi) +#R #L1 #L2 #l #i #H #Hli #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 +[ #H elim (ylt_yle_false … H Hli) | * /2 width=5 by ex3_2_intro/ ] qed-. -lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i → +lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i → ∀I,K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2. -#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 -[ #H elim (ylt_yle_false … H Hdi) +#R #L1 #L2 #l #i #H #Hli #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 +[ #H elim (ylt_yle_false … H Hli) | * /2 width=5 by ex3_2_intro/ ] qed-. -lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i → +lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i → ∀I1,I2,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2. -#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 -elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d +#R #L1 #L2 #l #i #HL12 #Hli #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 +elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -l #J #Y #HY lapply (drop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/ qed-. -fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 → - ∀K1,K2,I,V1,V2. ⬇[d] L1 ≡ K1.ⓑ{I}V1 → ⬇[d] L2 ≡ K2.ⓑ{I}V2 → - llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R d T L1 L2. -#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0 +fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,l0. llpx_sn R l0 T L1 L2 → ∀l. l0 = l + 1 → + ∀K1,K2,I,V1,V2. ⬇[l] L1 ≡ K1.ⓑ{I}V1 → ⬇[l] L2 ≡ K2.ⓑ{I}V2 → + llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2. +#R #L1 #L2 #T #l0 #H elim H -L1 -L2 -T -l0 /2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/ -[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct - elim (yle_split_eq i d) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hid +[ #L1 #L2 #l0 #i #HL12 #Hil #l #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct + elim (yle_split_eq i l) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hil #H destruct /2 width=9 by llpx_sn_lref/ -| #I #L1 #L2 #K11 #K22 #V1 #V2 #d0 #i #Hd0i #HLK11 #HLK22 #HK12 #HV12 #_ #d #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct +| #I #L1 #L2 #K11 #K22 #V1 #V2 #l0 #i #Hl0i #HLK11 #HLK22 #HK12 #HV12 #_ #l #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct /3 width=9 by llpx_sn_lref, yle_pred_sn/ -| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct +| #a #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct /4 width=9 by llpx_sn_bind, drop_drop/ -| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct +| #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct /3 width=9 by llpx_sn_flat/ ] qed-. -lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 → - ∀K1,K2,I,V1,V2. ⬇[d] L1 ≡ K1.ⓑ{I}V1 → ⬇[d] L2 ≡ K2.ⓑ{I}V2 → - llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R d T L1 L2. +lemma llpx_sn_inv_S: ∀R,L1,L2,T,l. llpx_sn R (l + 1) T L1 L2 → + ∀K1,K2,I,V1,V2. ⬇[l] L1 ≡ K1.ⓑ{I}V1 → ⬇[l] L2 ≡ K2.ⓑ{I}V2 → + llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2. /2 width=9 by llpx_sn_inv_S_aux/ qed-. lemma llpx_sn_inv_bind_O: ∀R. (∀L. reflexive … (R L)) → @@ -126,13 +126,13 @@ lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2 /3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-. lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2). + ∀T,L1,L2,l. Decidable (llpx_sn R l T L1 L2). #R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * * [ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/ | #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) - [ #HL12 #d elim (ylt_split i d) /3 width=1 by llpx_sn_skip, or_introl/ - #Hdi elim (lt_or_ge i (|L1|)) #HiL1 + [ #HL12 #l elim (ylt_split i l) /3 width=1 by llpx_sn_skip, or_introl/ + #Hli elim (lt_or_ge i (|L1|)) #HiL1 elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/ elim (drop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2 elim (drop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1 @@ -153,14 +153,14 @@ lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → #H #H0 destruct /2 width=1 by/ ] | #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/ -| #a #I #V #T #Hn #L2 #d destruct - elim (IH L1 V … L2 d) /2 width=1 by/ - elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯d)) -IH /3 width=1 by or_introl, llpx_sn_bind/ +| #a #I #V #T #Hn #L2 #l destruct + elim (IH L1 V … L2 l) /2 width=1 by/ + elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯l)) -IH /3 width=1 by or_introl, llpx_sn_bind/ #H1 #H2 @or_intror #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/ -| #I #V #T #Hn #L2 #d destruct - elim (IH L1 V … L2 d) /2 width=1 by/ - elim (IH L1 T … L2 d) -IH /3 width=1 by or_introl, llpx_sn_flat/ +| #I #V #T #Hn #L2 #l destruct + elim (IH L1 V … L2 l) /2 width=1 by/ + elim (IH L1 T … L2 l) -IH /3 width=1 by or_introl, llpx_sn_flat/ #H1 #H2 @or_intror #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/ ] @@ -169,115 +169,115 @@ qed-. (* Properties on relocation *************************************************) -lemma llpx_sn_lift_le: ∀R. l_liftable R → - ∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 → - ∀L1,L2,d,e. ⬇[Ⓕ, d, e] L1 ≡ K1 → ⬇[Ⓕ, d, e] L2 ≡ K2 → - ∀U. ⬆[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2. -#R #HR #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0 -[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d +lemma llpx_sn_lift_le: ∀R. d_liftable R → + ∀K1,K2,T,l0. llpx_sn R l0 T K1 K2 → + ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀U. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 U L1 L2. +#R #HR #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0 +[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l /2 width=1 by llpx_sn_sort/ -| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H - * #Hdi #H destruct - [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d +| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hli #H destruct + [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l /2 width=1 by llpx_sn_skip/ - | elim (ylt_yle_false … Hid0) -L1 -L2 -K1 -K2 -e -Hid0 + | elim (ylt_yle_false … Hil0) -L1 -L2 -K1 -K2 -m -Hil0 /3 width=3 by yle_trans, yle_inj/ ] -| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H - * #Hdi #H destruct [ -HK12 | -IHK12 ] +| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hli #H destruct [ -HK12 | -IHK12 ] [ elim (drop_trans_lt … HLK1 … HK11) // -K1 - elim (drop_trans_lt … HLK2 … HK22) // -Hdi -K2 + elim (drop_trans_lt … HLK2 … HK22) // -Hli -K2 /3 width=18 by llpx_sn_lref/ | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1 - lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hdi -Hd0 -K2 + lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hli -Hl0 -K2 /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/ ] -| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H - * #Hid #H destruct +| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hil #H destruct lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12 [ /3 width=7 by llpx_sn_free, drop_fwd_be/ | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1 lapply (drop_fwd_length … HLK2) -HLK2 #HLK2 - @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) + @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) ] -| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d -e +| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l -m /2 width=1 by llpx_sn_gref/ -| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H +| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/ -| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H +| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ ] qed-. -lemma llpx_sn_lift_ge: ∀R,K1,K2,T,d0. llpx_sn R d0 T K1 K2 → - ∀L1,L2,d,e. ⬇[Ⓕ, d, e] L1 ≡ K1 → ⬇[Ⓕ, d, e] L2 ≡ K2 → - ∀U. ⬆[d, e] T ≡ U → d ≤ d0 → llpx_sn R (d0+e) U L1 L2. -#R #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0 -[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d +lemma llpx_sn_lift_ge: ∀R,K1,K2,T,l0. llpx_sn R l0 T K1 K2 → + ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀U. ⬆[l, m] T ≡ U → l ≤ l0 → llpx_sn R (l0+m) U L1 L2. +#R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0 +[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l /2 width=1 by llpx_sn_sort/ -| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H +| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H * #_ #H destruct lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/ | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/ ] -| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H - * #Hid #H destruct - [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -e -Hid0 +| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hil #H destruct + [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -m -Hil0 /3 width=3 by ylt_yle_trans, ylt_inj/ | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1 - lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hid -Hd0 -K2 + lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hil -Hl0 -K2 /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/ ] -| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 … H) -H - * #Hid #H destruct +| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H + * #Hil #H destruct lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12 [ /3 width=7 by llpx_sn_free, drop_fwd_be/ | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1 lapply (drop_fwd_length … HLK2) -HLK2 #HLK2 - @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) + @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *) ] -| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X - lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -d +| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X + lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l /2 width=1 by llpx_sn_gref/ -| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 … H) -H +| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_skip, yle_succ/ -| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 … H) -H +| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/ ] qed-. (* Inversion lemmas on relocation *******************************************) -lemma llpx_sn_inv_lift_le: ∀R. l_deliftable_sn R → - ∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 → - ∀K1,K2,d,e. ⬇[Ⓕ, d, e] L1 ≡ K1 → ⬇[Ⓕ, d, e] L2 ≡ K2 → - ∀T. ⬆[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2. -#R #HR #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0 -[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e +lemma llpx_sn_inv_lift_le: ∀R. d_deliftable_sn R → + ∀L1,L2,U,l0. llpx_sn R l0 U L1 L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 T K1 K2. +#R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 +[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H +| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H * #_ #H destruct lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 [ /2 width=1 by llpx_sn_skip/ | /3 width=3 by llpx_sn_skip, yle_ylt_trans/ ] -| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H - * #Hid #H destruct [ -HK12 | -IHK12 ] +| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H + * #Hil #H destruct [ -HK12 | -IHK12 ] [ elim (drop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1 - elim (drop_conf_lt … HLK2 … HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2 + elim (drop_conf_lt … HLK2 … HLK22) // -Hil -L2 #L2 #V2 #HKL2 #HKL22 #HVW2 elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12 lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct /3 width=10 by llpx_sn_lref/ | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 - lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0 - elim (le_inv_plus_l … Hid) -Hid /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *) + lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 + elim (le_inv_plus_l … Hil) -Hil /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *) ] -| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 … H) -H +| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H * #_ #H destruct lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 @@ -287,39 +287,39 @@ lemma llpx_sn_inv_lift_le: ∀R. l_deliftable_sn R → lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ ] -| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d -e +| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind2 … H) -H +| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/ -| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat2 … H) -H +| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ ] qed-. -lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 → - ∀K1,K2,d,e. ⬇[Ⓕ, d, e] L1 ≡ K1 → ⬇[Ⓕ, d, e] L2 ≡ K2 → - ∀T. ⬆[d, e] T ≡ U → d ≤ d0 → d0 ≤ yinj d + e → llpx_sn R d T K1 K2. -#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0 -[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e +lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → l ≤ l0 → l0 ≤ yinj l + m → llpx_sn R l T K1 K2. +#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 +[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H - * #Hid #H destruct +| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H + * #Hil #H destruct [ lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 - -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/ - | elim (ylt_yle_false … Hid0) -L1 -L2 -Hd0 -Hid0 + -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/ + | elim (ylt_yle_false … Hil0) -L1 -L2 -Hl0 -Hil0 /3 width=3 by yle_trans, yle_inj/ (**) (* slow *) ] -| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H - * #Hid #H destruct - [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0 +| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H + * #Hil #H destruct + [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0m -Hil0 /3 width=3 by ylt_yle_trans, ylt_inj/ | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 - lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e - elim (le_inv_plus_l … Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/ + lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0m + elim (le_inv_plus_l … Hil) -Hil /3 width=9 by llpx_sn_lref, yle_inj/ ] -| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 … H) -H +| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H * #_ #H destruct lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 @@ -329,41 +329,41 @@ lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 → lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ ] -| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d0 -e +| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_bind2 … H) -H +| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_bind2 … H) -H >commutative_plus #V #T #HVW #HTU #H destruct @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *) @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/ -| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_flat2 … H) -H +| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/ ] qed-. -lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 → - ∀K1,K2,d,e. ⬇[Ⓕ, d, e] L1 ≡ K1 → ⬇[Ⓕ, d, e] L2 ≡ K2 → - ∀T. ⬆[d, e] T ≡ U → yinj d + e ≤ d0 → llpx_sn R (d0-e) T K1 K2. -#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0 -[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d +lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 → + ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 → + ∀T. ⬆[l, m] T ≡ U → yinj l + m ≤ l0 → llpx_sn R (l0-m) T K1 K2. +#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0 +[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l /2 width=1 by llpx_sn_sort/ -| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H - * #Hid #H destruct [ -Hid0 | -Hded0 ] +| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H + * #Hil #H destruct [ -Hil0 | -Hlml0 ] lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/ - | elim (le_inv_plus_l … Hid) -Hid #_ + | elim (le_inv_plus_l … Hil) -Hil #_ /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/ ] -| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H - * #Hid #H destruct - [ elim (ylt_yle_false … Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hid0 +| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H + * #Hil #H destruct + [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0 /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/ | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1 - lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hded0 -Hid + lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hlml0 -Hil /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/ ] -| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 … H) -H +| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H * #_ #H destruct lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) [ lapply (drop_fwd_length_le4 … HLK1) -HLK1 @@ -373,15 +373,15 @@ lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,d0. llpx_sn R d0 U L1 L2 → lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H /3 width=1 by llpx_sn_free, le_plus_to_minus_r/ ] -| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X - lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -d +| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X + lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l /2 width=1 by llpx_sn_gref/ -| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_bind2 … H) -H +| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)