X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpg_drops.ma;h=a98e061f39d5a61c872c36ca97cf219d573453f3;hp=779ea690b035a6c6b06e4298d1577f747f1be200;hb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;hpb=f694e3336cbdabdeefd86f85d827edfd26bf3464 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma index 779ea690b..a98e061f3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma @@ -12,48 +12,48 @@ (* *) (**************************************************************************) -include "basic_2/relocation/drops_drops.ma". -include "basic_2/s_computation/fqup_weight.ma". -include "basic_2/s_computation/fqup_drops.ma". +include "static_2/relocation/drops_drops.ma". +include "static_2/s_computation/fqup_weight.ma". +include "static_2/s_computation/fqup_drops.ma". include "basic_2/rt_transition/cpg.ma". -(* COUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************) +(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************) (* Advanced properties ******************************************************) -lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 → - ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[c, h] T2. -#c #h #G #K #V #V2 #i elim i -i +lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[Rt, c, h] V2 → + ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c, h] T2. +#Rt #c #h #G #K #V #V2 #i elim i -i [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/ | #i #IH #L0 #T0 #H0 #HV2 #HVT2 - elim (drops_inv_succ … H0) -H0 #I #L #V0 #HLK #H destruct - elim (lifts_split_trans … HVT2 (𝐔❴⫯i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/ + elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct + elim (lifts_split_trans … HVT2 (𝐔❴↑i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/ ] qed. -lemma cpg_ell_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 → - ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[(↓c)+𝟘𝟙, h] T2. -#c #h #G #K #V #V2 #i elim i -i +lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[Rt,c, h] V2 → + ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c+𝟘𝟙, h] T2. +#Rt #c #h #G #K #V #V2 #i elim i -i [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/ | #i #IH #L0 #T0 #H0 #HV2 #HVT2 - elim (drops_inv_succ … H0) -H0 #I #L #V0 #HLK #H destruct - elim (lifts_split_trans … HVT2 (𝐔❴⫯i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/ + elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct + elim (lifts_split_trans … HVT2 (𝐔❴↑i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/ ] qed. (* Advanced inversion lemmas ************************************************) -lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[c, h] T2 → +lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 → ∨∨ T2 = #i ∧ c = 𝟘𝟘 - | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 & - ⬆*[⫯i] V2 ≡ T2 & c = cV - | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 & - ⬆*[⫯i] V2 ≡ T2 & c = (↓cV) + 𝟘𝟙. -#c #h #G #i elim i -i + | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + ⬆*[↑i] V2 ≘ T2 & c = cV + | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + ⬆*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙. +#Rt #c #h #G #i elim i -i [ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/ /4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/ | #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/ - #I #K #V #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H + #I #K #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H [ * #H1 #H2 destruct lapply (lifts_inv_lref1_uni … HVT2) -HVT2 #H destruct /3 width=1 by or3_intro0, conj/ ] * #cV #L #W #W2 #HKL #HW2 #HWV2 #H destruct lapply (lifts_trans … HWV2 … HVT2 ??) -V2 @@ -61,14 +61,14 @@ lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[c, h] T2 → ] qed-. -lemma cpg_inv_atom1_drops: ∀c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[c, h] T2 → +lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, c, h] T2 → ∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘 | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙 - | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 & - ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = cV - | ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 & - ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = (↓cV) + 𝟘𝟙. -#c #h * #n #G #L #T2 #H + | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV + | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙. +#Rt #c #h * #n #G #L #T2 #H [ elim (cpg_inv_sort1 … H) -H * /3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/ | elim (cpg_inv_lref1_drops … H) -H * @@ -80,24 +80,27 @@ qed-. (* Properties with generic slicing for local environments *******************) -lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G). -#c #h #G #K #T generalize in match c; -c -@(fqup_wf_ind_eq … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * +(* Note: it should use drops_split_trans_pair2 *) +lemma cpg_lifts_sn: ∀Rt. reflexive … Rt → + ∀c,h,G. d_liftable2_sn … lifts (cpg Rt h c G). +#Rt #HRt #c #h #G #K #T generalize in match c; -c +@(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * [ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH lapply (lifts_inv_sort1 … H1) -H1 #H destruct elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/ | #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct elim (cpg_inv_lref1_drops … H2) -H2 * - [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ] + [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ] #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY - lapply (drops_inv_tls_at … Hf … HY) -HY #HY - elim (drops_inv_skip2 … HY) -HY #L0 #W #HLK0 #HVW #H destruct + lapply (drops_tls_at … Hf … HY) -HY #HY + elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct + elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2 - elim (lifts_total W2 (𝐔❴⫯i2❵)) #U2 #HWU2 + elim (lifts_total W2 (𝐔❴↑i2❵)) #U2 #HWU2 lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2 lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ] /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/ @@ -110,64 +113,77 @@ lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G). elim (cpg_inv_bind1 … H2) -H2 * [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct elim (IH … HV12 … HLK … HVW1) -HV12 // - elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] + elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] /3 width=5 by cpg_bind, lifts_bind, ex2_intro/ | #cT #T2 #HT12 #HXT2 #H1 #H2 #H3 destruct - elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #U2 #HTU2 #HU12 + elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] #U2 #HTU2 #HU12 lapply (lifts_trans … HXT2 … HTU2 ??) -T2 [3: |*: // ] #HXU2 - elim (lifts_split_trans … HXU2 f (𝐔❴⫯O❵)) [2: /2 width=1 by after_uni_one_dx/ ] + elim (lifts_split_trans … HXU2 f (𝐔❴↑O❵)) [2: /2 width=1 by after_uni_one_dx/ ] /3 width=5 by cpg_zeta, ex2_intro/ ] -| #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct +| * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct - elim (cpg_inv_flat1 … H2) -H2 * - [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct - elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // - elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 // - /3 width=5 by cpg_flat, lifts_flat, ex2_intro/ - | #cT #HT12 #H1 #H2 destruct - elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 // - /3 width=3 by cpg_eps, ex2_intro/ - | #cV #HV12 #H1 #H2 destruct - elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 // - /3 width=3 by cpg_ee, ex2_intro/ - | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct - elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct - elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // - elim (IH … HY12 … HLK … HYZ1) -HY12 // - elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] - /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/ - | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct - elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct - elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12 - elim (IH … HY12 … HLK … HYZ1) -HY12 // - elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] - elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20 - lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H - lapply (lifts_conf … HV20 … H (↑f) ?) -V2 /2 width=3 by after_uni_one_sn/ - /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/ + [ elim (cpg_inv_appl1 … H2) -H2 * + [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct + elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // + elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 // + /3 width=5 by cpg_appl, lifts_flat, ex2_intro/ + | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 destruct + elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct + elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // + elim (IH … HY12 … HLK … HYZ1) -HY12 // + elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] + /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/ + | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 destruct + elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct + elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12 + elim (IH … HY12 … HLK … HYZ1) -HY12 // + elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] + elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20 + lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H + lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/ + /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/ + ] + | elim (cpg_inv_cast1 … H2) -H2 * + [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct + elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // + elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 // + /3 width=5 by cpg_cast, lifts_flat, ex2_intro/ + | #cT #HT12 #H destruct + elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 // + /3 width=3 by cpg_eps, ex2_intro/ + | #cV #HV12 #H destruct + elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 // + /3 width=3 by cpg_ee, ex2_intro/ + ] ] ] qed-. +lemma cpg_lifts_bi: ∀Rt. reflexive … Rt → + ∀c,h,G. d_liftable2_bi … lifts (cpg Rt h c G). +/3 width=12 by cpg_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-. + (* Inversion lemmas with generic slicing for local environments *************) -lemma cpg_inv_lifts1: ∀c,h,G. d_deliftable2_sn (cpg h c G). -#c #h #G #L #U generalize in match c; -c -@(fqup_wf_ind_eq … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * * +lemma cpg_inv_lifts_sn: ∀Rt. reflexive … Rt → + ∀c,h,G. d_deliftable2_sn … lifts (cpg Rt h c G). +#Rt #HRt #c #h #G #L #U generalize in match c; -c +@(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * * [ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH lapply (lifts_inv_sort2 … H1) -H1 #H destruct elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/ | #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct elim (cpg_inv_lref1_drops … H2) -H2 * - [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ] + [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ] #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0 lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0 - lapply (drops_inv_tls_at … Hf … HLY0) -HLY0 #HLY0 - elim (drops_inv_skip1 … HLY0) -HLY0 #K0 #V #HLK0 #HVW #H destruct + lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0 + elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct + elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2 lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2 elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ] @@ -181,39 +197,50 @@ lemma cpg_inv_lifts1: ∀c,h,G. d_deliftable2_sn (cpg h c G). elim (cpg_inv_bind1 … H2) -H2 * [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct elim (IH … HW12 … HLK … HVW1) -HW12 // - elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] + elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] /3 width=5 by cpg_bind, lifts_bind, ex2_intro/ | #cU #U2 #HU12 #HXU2 #H1 #H2 #H3 destruct - elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #T2 #HTU2 #HT12 + elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] #T2 #HTU2 #HT12 elim (lifts_div4_one … HTU2 … HXU2) -U2 /3 width=5 by cpg_zeta, ex2_intro/ ] -| #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct +| * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct - elim (cpg_inv_flat1 … H2) -H2 * - [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct - elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // - elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 // - /3 width=5 by cpg_flat, lifts_flat, ex2_intro/ - | #cU #HU12 #H1 #H2 destruct - elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 // - /3 width=3 by cpg_eps, ex2_intro/ - | #cW #HW12 #H1 #H2 destruct - elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 // - /3 width=3 by cpg_ee, ex2_intro/ - | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct - elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct - elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // - elim (IH … HZ12 … HLK … HYZ1) -HZ12 // - elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] - /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/ - | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct - elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct - elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12 - elim (IH … HZ12 … HLK … HYZ1) -HZ12 // - elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] - lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H - elim (lifts_split_trans … H ? (↑f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ] - /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/ + [ elim (cpg_inv_appl1 … H2) -H2 * + [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct + elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // + elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 // + /3 width=5 by cpg_appl, lifts_flat, ex2_intro/ + | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct + elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct + elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // + elim (IH … HZ12 … HLK … HYZ1) -HZ12 // + elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] + /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/ + | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct + elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct + elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12 + elim (IH … HZ12 … HLK … HYZ1) -HZ12 // + elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] + lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H + elim (lifts_split_trans … H ? (⫯f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ] + /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/ + ] + | elim (cpg_inv_cast1 … H2) -H2 * + [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct + elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // + elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 // + /3 width=5 by cpg_cast, lifts_flat, ex2_intro/ + | #cU #HU12 #H destruct + elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 // + /3 width=3 by cpg_eps, ex2_intro/ + | #cW #HW12 #H destruct + elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 // + /3 width=3 by cpg_ee, ex2_intro/ + ] ] ] qed-. + +lemma cpg_inv_lifts_bi: ∀Rt. reflexive … Rt → + ∀c,h,G. d_deliftable2_bi … lifts (cpg Rt h c G). +/3 width=12 by cpg_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.