(* Advanced properties ******************************************************)
-lemma fsle_lifts_sn: ∀T1,U1. ⇧*[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| →
- â\88\80T2. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84 â\86\92 â¦\83L1.â\93§,U1â¦\84 â\8a\86 â¦\83L2,T2â¦\84.
+lemma fsle_lifts_sn: ∀T1,U1. ⇧[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| →
+ â\88\80T2. â\9d¨L1,T1â\9d© â\8a\86 â\9d¨L2,T2â\9d© â\86\92 â\9d¨L1.â\93§,U1â\9d© â\8a\86 â\9d¨L2,T2â\9d©.
#T1 #U1 #HTU1 #L1 #L2 #H1L #T2
* #n #m #f #g #Hf #Hg #H2L #Hfg
lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct
qed-.
lemma fsle_lifts_dx (L1) (L2):
- |L1| ≤ |L2| → ∀T2,U2. ⇧*[1]T2 ≘ U2 →
- â\88\80T1. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84 â\86\92 â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2.â\93§,U2â¦\84.
+ |L1| ≤ |L2| → ∀T2,U2. ⇧[1]T2 ≘ U2 →
+ â\88\80T1. â\9d¨L1,T1â\9d© â\8a\86 â\9d¨L2,T2â\9d© â\86\92 â\9d¨L1,T1â\9d© â\8a\86 â\9d¨L2.â\93§,U2â\9d©.
#L1 #L2 #HL21 #T2 #U2 #HTU2 #T1
* #n #m #f #g #Hf #Hg #H2L #Hfg
lapply (lveq_length_fwd_sn … H2L ?) // -HL21 #H destruct
@(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_dx/ (**) (* explict constructor *)
qed-.
-lemma fsle_lifts_SO_sn: â\88\80K1,K2. |K1| = |K2| â\86\92 â\88\80V1,V2. â¦\83K1,V1â¦\84 â\8a\86 â¦\83K2,V2â¦\84 →
- ∀W1. ⇧*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1},W1⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄.
+lemma fsle_lifts_SO_sn: â\88\80K1,K2. |K1| = |K2| â\86\92 â\88\80V1,V2. â\9d¨K1,V1â\9d© â\8a\86 â\9d¨K2,V2â\9d© →
+ ∀W1. ⇧[1] V1 ≘ W1 → ∀I1,I2. ❨K1.ⓘ[I1],W1❩ ⊆ ❨K2.ⓑ[I2]V2,#O❩.
#K1 #K2 #HK #V1 #V2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#W1 #HVW1 #I1 #I2
elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
-/5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, sle_weak, ex4_4_intro/
+/5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, pr_sle_weak, ex4_4_intro/
qed.
-lemma fsle_lifts_SO: â\88\80K1,K2. |K1| = |K2| â\86\92 â\88\80T1,T2. â¦\83K1,T1â¦\84 â\8a\86 â¦\83K2,T2â¦\84 →
- ∀U1,U2. ⇧*[1] T1 ≘ U1 → ⇧*[1] T2 ≘ U2 →
- â\88\80I1,I2. â¦\83K1.â\93\98{I1},U1â¦\84 â\8a\86 â¦\83K2.â\93\98{I2},U2â¦\84.
+lemma fsle_lifts_SO: â\88\80K1,K2. |K1| = |K2| â\86\92 â\88\80T1,T2. â\9d¨K1,T1â\9d© â\8a\86 â\9d¨K2,T2â\9d© →
+ ∀U1,U2. ⇧[1] T1 ≘ U1 → ⇧[1] T2 ≘ U2 →
+ â\88\80I1,I2. â\9d¨K1.â\93\98[I1],U1â\9d© â\8a\86 â\9d¨K2.â\93\98[I2],U2â\9d©.
#K1 #K2 #HK #T1 #T2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#U1 #U2 #HTU1 #HTU2 #I1 #I2
elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
-/5 width=12 by frees_lifts_SO, drops_refl, drops_drop, lveq_bind, sle_push, ex4_4_intro/
+/5 width=12 by frees_lifts_SO, drops_refl, drops_drop, lveq_bind, pr_sle_push, ex4_4_intro/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma fsle_inv_lifts_sn: ∀T1,U1. ⇧*[1] T1 ≘ U1 →
- â\88\80I1,I2,L1,L2,V1,V2,U2. â¦\83L1.â\93\91{I1}V1,U1â¦\84 â\8a\86 â¦\83L2.â\93\91{I2}V2,U2â¦\84 →
- â\88\80p. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,â\93\91{p,I2}V2.U2â¦\84.
+lemma fsle_inv_lifts_sn: ∀T1,U1. ⇧[1] T1 ≘ U1 →
+ â\88\80I1,I2,L1,L2,V1,V2,U2. â\9d¨L1.â\93\91[I1]V1,U1â\9d© â\8a\86 â\9d¨L2.â\93\91[I2]V2,U2â\9d© →
+ â\88\80p. â\9d¨L1,T1â\9d© â\8a\86 â\9d¨L2,â\93\91[p,I2]V2.U2â\9d©.
#T1 #U1 #HTU1 #I1 #I2 #L1 #L2 #V1 #V2 #U2
* #n #m #f2 #g2 #Hf2 #Hg2 #HL #Hfg2 #p
elim (lveq_inv_pair_pair … HL) -HL #HL #H1 #H2 destruct
elim (frees_total L2 V2) #g1 #Hg1
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+elim (pr_sor_isf_bi g1 (⫰g2)) /3 width=3 by frees_fwd_isfin, pr_isf_tl/ #g #Hg #_
lapply (frees_inv_lifts_SO (Ⓣ) … Hf2 … HTU1)
[1,2: /3 width=4 by drops_refl, drops_drop/ ] -U1 #Hf2
-lapply (sor_inv_sle_dx … Hg) #H0g
-/5 width=10 by frees_bind, sle_tl, sle_trans, ex4_4_intro/
+lapply (pr_sor_inv_sle_dx … Hg) #H0g
+/5 width=10 by frees_bind, pr_sle_tl, pr_sle_trans, ex4_4_intro/
qed-.