(* Basic eliminators ********************************************************)
lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L2 T2.
+ (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄ → R L2 T2.
/3 width=7 by bi_TC_star_ind/ qed-.
lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L1 T1.
+ (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄ → R L1 T1.
/3 width=7 by bi_TC_star_ind_dx/ qed-.
(* Basic properties *********************************************************)
lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g).
/2 width=1/ qed.
-lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄ →
+ h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
/2 width=1/ qed.
-lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
/2 width=4/ qed-.
-lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
/2 width=4/ qed-.
(* Note: this is a general property of bi_TC *)
lemma fsupp_yprs: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+ h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
#h #g #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -L2 -T2 /3 width=1/ /3 width=4/
qed.
-lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
+lemma cprs_yprs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[h, g] ⦃L, T2⦄.
#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4 by ypr_cpr, yprs_strap1/
qed.
-lemma lprs_yprs: ∀h,g,L1,L2,T. L1 ⊢ ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
+lemma lprs_yprs: ∀h,g,L1,L2,T. L1 ⊢ ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[h, g] ⦃L2, T⦄.
#h #g #L1 #L2 #T #H @(lprs_ind … H) -L2 // /3 width=4 by ypr_lpr, yprs_strap1/
qed.
-lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
- h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
+lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 →
+ h ⊢ ⦃L, T1⦄ ≥[h, g] ⦃L, T2⦄.
#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=4 by ypr_ssta, yprs_strap1/
qed.
-lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
+lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[h, g] L1 → h ⊢ ⦃L1, T⦄ ≥[h, g] ⦃L2, T⦄.
/3 width=1/ qed.
lemma cprs_lpr_yprs: ∀h,g,L1,L2,T1,T2. L1 ⊢ T1 ➡* T2 → L1 ⊢ ➡ L2 →
- h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+ h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
/3 width=4 by yprs_strap1, ypr_lpr, cprs_yprs/
qed.