X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Fdrops_sex.ma;h=18017d88ae41fcce0471faaf4616ddc7d0a98620;hb=98e786e1a6bd7b621e37ba7cd4098d4a0a6f8278;hp=edec3dcc5f88841327da1a83856e5675629cfa05;hpb=bd53c4e895203eb049e75434f638f26b5a161a2b;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma index edec3dcc5..18017d88a 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma @@ -20,40 +20,42 @@ include "static_2/relocation/drops.ma". (* Properties with entrywise extension of context-sensitive relations *******) (**) (* changed after commit 13218 *) -lemma sex_co_dropable_sn: ∀RN,RP. co_dropable_sn (sex RN RP). +lemma sex_co_dropable_sn (RN) (RP): + co_dropable_sn (sex RN RP). #RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1 [ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(sex_inv_atom1 … H) -X /4 width=3 by sex_atom, drops_atom, ex2_intro/ | #f #I1 #L1 #K1 #_ #IH #Hf #f2 #X #H #f1 #Hf2 - elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct + elim (pr_coafter_inv_next_sn … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct elim (sex_inv_push1 … H) -H #I2 #L2 #HL12 #HI12 #H destruct elim (IH … HL12 … Hg2) -g2 - /3 width=3 by isuni_inv_next, drops_drop, ex2_intro/ + /3 width=3 by pr_isu_inv_next, drops_drop, ex2_intro/ | #f #I1 #J1 #L1 #K1 #HLK #HJI1 #IH #Hf #f2 #X #H #f1 #Hf2 - lapply (isuni_inv_push … Hf ??) -Hf [3: |*: // ] #Hf + lapply (pr_isu_inv_push … Hf ??) -Hf [3: |*: // ] #Hf lapply (drops_fwd_isid … HLK … Hf) -HLK #H0 destruct lapply (liftsb_fwd_isid … HJI1 … Hf) -HJI1 #H0 destruct - elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct + elim (pr_coafter_inv_push_sn … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct [ elim (sex_inv_push1 … H) | elim (sex_inv_next1 … H) ] -H #I2 #L2 #HL12 #HI12 #H destruct - elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by isuni_isid/ #K2 #HK12 #HLK2 + elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by pr_isu_isi/ #K2 #HK12 #HLK2 lapply (drops_fwd_isid … HLK2 … Hf) -HLK2 #H0 destruct - /4 width=3 by drops_refl, sex_next, sex_push, isid_push, ex2_intro/ + /4 width=3 by drops_refl, sex_next, sex_push, pr_isi_push, ex2_intro/ ] qed-. -lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → - ∀b,f. ⇩*[b,f] L1 ≘ K1 → ⇩*[b,f] L2 ≘ K2 → - f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2. +lemma sex_liftable_co_dedropable_bi (RN) (RP): + d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → + ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → + ∀b,f. ⇩*[b,f] L1 ≘ K1 → ⇩*[b,f] L2 ≘ K2 → + f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2. #RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 // #g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H -[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct +[ elim (pr_coafter_inv_next … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct elim (sex_inv_next … HK12) -HK12 #HK12 #HJ12 elim (HRN … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by sex_next/ -| elim (coafter_inv_xxp … H) [1,2: |*: // ] -H * +| elim (pr_coafter_inv_push … H) [1,2: |*: // ] -H * [ #g #g1 #Hg2 #H1 #H2 destruct elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct @@ -68,18 +70,19 @@ lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → ] qed-. -lemma sex_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) → - d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - co_dedropable_sn (sex RN RP). +lemma sex_liftable_co_dedropable_sn (RN) (RP): + (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) → + d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → + co_dedropable_sn (sex RN RP). #RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1 [ #f #Hf #X #f1 #H #f2 #Hf2 >(sex_inv_atom1 … H) -X /4 width=4 by drops_atom, sex_atom, ex3_intro/ | #f #I1 #L1 #K1 #_ #IHLK1 #K2 #f1 #HK12 #f2 #Hf2 - elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct + elim (pr_coafter_inv_next_sn … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct elim (IHLK1 … HK12 … Hg2) -K1 /3 width=6 by drops_drop, sex_next, sex_push, ex3_intro/ | #f #I1 #J1 #L1 #K1 #HLK1 #HJI1 #IHLK1 #X #f1 #H #f2 #Hf2 - elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct + elim (pr_coafter_inv_push_sn … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct [ elim (sex_inv_push1 … H) | elim (sex_inv_next1 … H) ] -H #J2 #K2 #HK12 #HJ12 #H destruct [ elim (H2RP … HJ12 … HLK1 … HJI1) | elim (H2RN … HJ12 … HLK1 … HJI1) ] -J1 elim (IHLK1 … HK12 … Hg2) -K1 @@ -87,101 +90,135 @@ lemma sex_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → ] qed-. -fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔❪f❫ → - ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 → - ∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2. +fact sex_dropable_dx_aux (RN) (RP): + ∀b,f,L2,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔❪f❫ → + ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 → + ∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2. #RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2 [ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (sex_inv_atom2 … H) -H #H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/ | #f #I2 #L2 #K2 #_ #IH #Hf #f2 #X #HX #f1 #Hf2 - elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct + elim (pr_coafter_inv_next_sn … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct elim (sex_inv_push2 … HX) -HX #I1 #L1 #HL12 #HI12 #H destruct elim (IH … HL12 … Hg2) -L2 -I2 -g2 - /3 width=3 by drops_drop, isuni_inv_next, ex2_intro/ + /3 width=3 by drops_drop, pr_isu_inv_next, ex2_intro/ | #f #I2 #J2 #L2 #K2 #_ #HJI2 #IH #Hf #f2 #X #HX #f1 #Hf2 - elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct + elim (pr_coafter_inv_push_sn … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct [ elim (sex_inv_push2 … HX) | elim (sex_inv_next2 … HX) ] -HX #I1 #L1 #HL12 #HI12 #H destruct - elim (IH … HL12 … Hg2) -L2 -g2 /2 width=3 by isuni_fwd_push/ #K1 #HLK1 #HK12 - lapply (isuni_inv_push … Hf ??) -Hf [3,6: |*: // ] #Hf + elim (IH … HL12 … Hg2) -L2 -g2 /2 width=3 by pr_isu_fwd_push/ #K1 #HLK1 #HK12 + lapply (pr_isu_inv_push … Hf ??) -Hf [3,6: |*: // ] #Hf lapply (liftsb_fwd_isid … HJI2 … Hf) #H destruct -HJI2 lapply (drops_fwd_isid … HLK1 … Hf) #H destruct -HLK1 - /4 width=5 by sex_next, sex_push, drops_refl, isid_push, ex2_intro/ + /4 width=5 by sex_next, sex_push, drops_refl, pr_isi_push, ex2_intro/ ] qed-. -lemma sex_co_dropable_dx: ∀RN,RP. co_dropable_dx (sex RN RP). +lemma sex_co_dropable_dx (RN) (RP): + co_dropable_dx (sex RN RP). /2 width=5 by sex_dropable_dx_aux/ qed-. -lemma sex_drops_conf_next: ∀RN,RP. - ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → - ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ → - ∀f1. f ~⊚ ↑f1 ≘ f2 → - ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2. +lemma sex_drops_conf_next (RN) (RP): + ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ → + ∀f1. f ~⊚ ↑f1 ≘ f2 → + ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2 elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf #X #HX #HLK2 elim (sex_inv_next1 … HX) -HX #I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma sex_drops_conf_push: ∀RN,RP. - ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → - ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ → - ∀f1. f ~⊚ ⫯f1 ≘ f2 → - ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2. +lemma sex_drops_conf_push (RN) (RP): + ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ → + ∀f1. f ~⊚ ⫯f1 ≘ f2 → + ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2 elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf #X #HX #HLK2 elim (sex_inv_push1 … HX) -HX #I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → - ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ → - ∀f1. f ~⊚ ↑f1 ≘ f2 → - ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2. +lemma sex_drops_trans_next (RN) (RP): + ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ → + ∀f1. f ~⊚ ↑f1 ≘ f2 → + ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2 elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf #X #HLK1 #HX elim (sex_inv_next2 … HX) -HX #I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → - ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ → - ∀f1. f ~⊚ ⫯f1 ≘ f2 → - ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2. +lemma sex_drops_trans_push (RN) (RP): ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ → + ∀f1. f ~⊚ ⫯f1 ≘ f2 → + ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2 elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf #X #HLK1 #HX elim (sex_inv_push2 … HX) -HX #I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → - d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → - ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 → - ∀f2. f ~⊚ f1 ≘ ↑f2 → - ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2]. +lemma drops_sex_trans_next (RN) (RP): + (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → + d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → + ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → + ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 → + ∀f2. f ~⊚ f1 ≘ ↑f2 → + ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2]. #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2 elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP #X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX #I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/ qed-. -lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → - d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → - ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 → - ∀f2. f ~⊚ f1 ≘ ⫯f2 → - ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2]. +lemma drops_sex_trans_push (RN) (RP): + (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → + d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → + ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → + ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 → + ∀f2. f ~⊚ f1 ≘ ⫯f2 → + ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2]. #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2 elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP #X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX #I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/ qed-. -lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⇩*[b,f1] L1 ≘ ⋆ → 𝐔❪f1❫ → - ∀f,L2. L1 ⪤[RN,RP,f] L2 → - ∀f2. f1 ~⊚ f2 ≘f → ⇩*[b,f1] L2 ≘ ⋆. +lemma drops_atom2_sex_conf (RN) (RP): + ∀b,f1,L1. ⇩*[b,f1] L1 ≘ ⋆ → 𝐔❪f1❫ → + ∀f,L2. L1 ⪤[RN,RP,f] L2 → + ∀f2. f1 ~⊚ f2 ≘f → ⇩*[b,f1] L2 ≘ ⋆. #RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3 elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1 #L #H #HL2 lapply (sex_inv_atom1 … H) -H // qed-. + +lemma sex_sdj_split_dx (R1) (R2) (RP): + c_reflexive … R1 → c_reflexive … R2 → c_reflexive … RP → + ∀L1,f1. + (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ↑g = ⫰*[n] f1 → R_pw_confluent1_sex R1 R1 R2 cfull g K I) → + ∀L2. L1 ⪤[R1,RP,f1] L2 → ∀f2. f1 ∥ f2 → + ∃∃L. L1 ⪤[R2,cfull,f1] L & L ⪤[RP,R1,f2] L2. +#R1 #R2 #RP #HR1 #HR2 #HRP #L1 elim L1 -L1 +[ #f1 #_ #L2 #H #f2 #_ + lapply (sex_inv_atom1 … H) -H #H destruct + /2 width=3 by sex_atom, ex2_intro/ +| #K1 #I1 #IH #f1 elim (pr_map_split_tl f1) * #g1 #H destruct + #HR #L2 #H #f2 #Hf + [ elim (sex_inv_push1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct + elim (pr_sdj_inv_push_sn … Hf ??) -Hf [1,3: * |*: // ] #g2 #Hg #H destruct + elim (IH … HK12 … Hg) -IH -HK12 -Hg + [1,3: /3 width=5 by sex_next, sex_push, ex2_intro/ + |2,4: /3 width=3 by drops_drop/ + ] + | elim (sex_inv_next1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct + elim (pr_sdj_inv_next_sn … Hf ??) -Hf [|*: // ] #g2 #Hg #H destruct + elim (IH … HK12 … Hg) -IH -HK12 -Hg + [ /5 width=11 by sex_next, sex_push, drops_refl, ex2_intro/ + | /3 width=3 by drops_drop/ + ] + ] +] +qed-.