| lveq_bind : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 →
lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓘ{I2})
| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 0 K2 →
- lveq (⫯n1) (K1.ⓧ) 0 K2
+ lveq (â\86\91n1) (K1.ⓧ) 0 K2
| lveq_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 →
- lveq 0 K1 (⫯n2) (K2.ⓧ)
+ lveq 0 K1 (â\86\91n2) (K2.ⓧ)
.
interpretation "equivalence up to exclusion binders (local environment)"
/2 width=5 by lveq_inv_zero_aux/ qed-.
fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- â\88\80m1. ⫯m1 = n1 →
+ â\88\80m1. â\86\91m1 = n1 →
∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: #m #H destruct
]
qed-.
-lemma lveq_inv_succ_sn: â\88\80L1,K2,n1,n2. L1 â\89\8bâ\93§*[⫯n1, n2] K2 →
+lemma lveq_inv_succ_sn: â\88\80L1,K2,n1,n2. L1 â\89\8bâ\93§*[â\86\91n1, n2] K2 →
∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2.
/2 width=3 by lveq_inv_succ_sn_aux/ qed-.
-lemma lveq_inv_succ_dx: â\88\80K1,L2,n1,n2. K1 â\89\8bâ\93§*[n1, ⫯n2] L2 →
+lemma lveq_inv_succ_dx: â\88\80K1,L2,n1,n2. K1 â\89\8bâ\93§*[n1, â\86\91n2] L2 →
∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1.
#K1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
qed-.
fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- â\88\80m1,m2. ⫯m1 = n1 â\86\92 ⫯m2 = n2 → ⊥.
+ â\88\80m1,m2. â\86\91m1 = n1 â\86\92 â\86\91m2 = n2 → ⊥.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: #m1 #m2 #H1 #H2 destruct
|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_succ: â\88\80L1,L2,n1,n2. L1 â\89\8bâ\93§*[⫯n1, ⫯n2] L2 → ⊥.
+lemma lveq_inv_succ: â\88\80L1,L2,n1,n2. L1 â\89\8bâ\93§*[â\86\91n1, â\86\91n2] L2 → ⊥.
/2 width=9 by lveq_inv_succ_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
qed-.
lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
- â\88\83â\88\83m1. K1 â\89\8bâ\93§*[m1, 0] â\8b\86 & BUnit Void = I1 & ⫯m1 = n1 & 0 = n2.
+ â\88\83â\88\83m1. K1 â\89\8bâ\93§*[m1, 0] â\8b\86 & BUnit Void = I1 & â\86\91m1 = n1 & 0 = n2.
#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
qed-.
lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} →
- â\88\83â\88\83m2. â\8b\86 â\89\8bâ\93§*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ⫯m2 = n2.
+ â\88\83â\88\83m2. â\8b\86 â\89\8bâ\93§*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & â\86\91m2 = n2.
#I2 #K2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_bind_atom … H) -H
]
qed-.
-lemma lveq_inv_void_succ_sn: â\88\80L1,L2,n1,n2. L1.â\93§ â\89\8bâ\93§*[⫯n1, n2] L2 →
+lemma lveq_inv_void_succ_sn: â\88\80L1,L2,n1,n2. L1.â\93§ â\89\8bâ\93§*[â\86\91n1, n2] L2 →
∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2.
#L1 #L2 #n1 #n2 #H
elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
qed-.
-lemma lveq_inv_void_succ_dx: â\88\80L1,L2,n1,n2. L1 â\89\8bâ\93§*[n1, ⫯n2] L2.ⓧ →
+lemma lveq_inv_void_succ_dx: â\88\80L1,L2,n1,n2. L1 â\89\8bâ\93§*[n1, â\86\91n2] L2.ⓧ →
∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1.
#L1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H