inductive lveq: bi_relation nat lenv ≝
| lveq_atom : lveq 0 (⋆) 0 (⋆)
-| lveq_pair_sn: ∀I1,I2,K1,K2,V1,n. lveq n K1 n K2 →
- lveq 0 (K1.ⓑ{I1}V1) 0 (K2.ⓘ{I2})
-| lveq_pair_dx: ∀I1,I2,K1,K2,V2,n. lveq n K1 n K2 →
- lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓑ{I2}V2)
-| lveq_void_sn: ∀K1,K2,n1,n2. lveq n1 K1 n2 K2 →
- lveq (⫯n1) (K1.ⓧ) n2 K2
-| lveq_void_dx: ∀K1,K2,n1,n2. lveq n1 K1 n2 K2 →
- lveq n1 K1 (⫯n2) (K2.ⓧ)
+| 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_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 →
+ lveq 0 K1 (↑n2) (K2.ⓧ)
.
interpretation "equivalence up to exclusion binders (local environment)"
(* Basic properties *********************************************************)
-lemma lveq_refl: ∀L. ∃n. L ≋ⓧ*[n, n] L.
-#L elim L -L /2 width=2 by ex_intro, lveq_atom/
-#L #I * #n #IH cases I -I /3 width=2 by ex_intro, lveq_pair_dx/
-* /4 width=2 by ex_intro, lveq_void_sn, lveq_void_dx/
-qed-.
+lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L.
+#L elim L -L /2 width=1 by lveq_atom, lveq_bind/
+qed.
lemma lveq_sym: bi_symmetric … lveq.
#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
-/2 width=2 by lveq_atom, lveq_pair_sn, lveq_pair_dx, lveq_void_sn, lveq_void_dx/
+/2 width=1 by lveq_atom, lveq_bind, lveq_void_sn, lveq_void_dx/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact lveq_inv_atom_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ⋆ = L1 → ⋆ = L2 → ∧∧ 0 = n1 & 0 = n2.
+fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ 0 = n1 → 0 = n2 →
+ ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
+[1: /3 width=1 by or_introl, conj/
+|2: /3 width=7 by ex3_4_intro, or_intror/
+|*: #K1 #K2 #n #_ #H1 #H2 destruct
+]
+qed-.
+
+lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
+ ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+/2 width=5 by lveq_inv_zero_aux/ qed-.
+
+fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∀m1. ↑m1 = n1 →
+ ∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2.
+#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
+[1: #m #H destruct
+|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
+|*: #K1 #K2 #n #HK #m #H destruct /2 width=3 by ex3_intro/
+]
+qed-.
+
+lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, 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: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1, ↑n2] L2 →
+ ∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1.
+#K1 #L2 #n1 #n2 #H
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
+qed-.
+
+fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[ /2 width=1 by conj/
-|2,3: #I1 #I2 #K1 #K2 #V #n #_ #H1 #H2 destruct
-|4,5: #K1 #K2 #n1 #n2 #_ #H1 #H2 destruct
+[1: #m1 #m2 #H1 #H2 destruct
+|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct
+|*: #K1 #K2 #n #_ #m1 #m2 #H1 #H2 destruct
]
qed-.
+lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥.
+/2 width=9 by lveq_inv_succ_aux/ qed-.
+
(* Advanced inversion lemmas ************************************************)
-lemma lveq_inv_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → 0 = n1 ∧ 0 = n2.
-/2 width=5 by lveq_inv_atom_aux/ qed-.
-
-fact lveq_inv_void_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀K1,m1. L1 = K1.ⓧ → n1 = ⫯m1 → K1 ≋ ⓧ*[m1, n2] L2.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2
-[ #K2 #m2 #H destruct
-| #I1 #I2 #L1 #L2 #V #n #_ #_ #K1 #m1 #H1 #H2 destruct
-| #I1 #I2 #L1 #L2 #V #n #_ #_ #K1 #m1 #H1 #H2 destruct
-| #L1 #L2 #n1 #n2 #HL12 #_ #K1 #m1 #H1 #H2 destruct //
-| #L1 #L2 #n1 #n2 #_ #IH #K1 #m1 #H1 #H2 destruct
- /3 width=1 by lveq_void_dx/
+lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2.
+#I1 #I2 #K1 #K2 #H
+elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
+qed-.
+
+lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
+* [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
+| /2 width=1 by conj/
]
qed-.
-lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[⫯n1, n2] L2 → L1 ≋ ⓧ*[n1, n2] L2.
-/2 width=5 by lveq_inv_void_succ_sn_aux/ qed-.
+lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
+ ∃∃m1. K1 ≋ⓧ*[m1, 0] ⋆ & BUnit Void = I1 & ↑m1 = 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
+| elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=3 by ex4_intro/
+| elim (lveq_inv_zero … H) -H *
+ [ #H1 #H2 destruct
+ | #Z1 #Z2 #Y1 #Y2 #_ #H1 #H2 destruct
+ ]
+]
+qed-.
-lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ⫯n2] L2.ⓧ → L1 ≋ ⓧ*[n1, n2] L2.
-/4 width=5 by lveq_inv_void_succ_sn_aux, lveq_sym/ qed-.
+lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} →
+ ∃∃m2. ⋆ ≋ⓧ*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
+#I2 #K2 #n1 #n2 #H
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_bind_atom … H) -H
+/3 width=3 by lveq_sym, ex4_intro/
+qed-.
+
+lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 →
+ ∧∧ K1 ≋ⓧ*[0, 0] K2 & 0 = n1 & 0 = n2.
+#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_zero … H) -H *
+ [ #H1 #H2 destruct
+ | #Z1 #Z2 #Y1 #Y2 #HY #H1 #H2 destruct /3 width=1 by and3_intro/
+ ]
+]
+qed-.
+
+lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, 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: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ↑n2] L2.ⓧ →
+ ∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1.
+#L1 #L2 #n1 #n2 #H
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_void_succ_sn … H) -H
+/3 width=1 by lveq_sym, conj/
+qed-.
(* Advanced forward lemmas **************************************************)
-fact lveq_fwd_pair_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀I,K1,V. K1.ⓑ{I}V = L1 → 0 = n1.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 //
-#K1 #K2 #n1 #n2 #_ #IH #J #L1 #V #H destruct /2 width=4 by/
+lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∨∨ 0 = n1 | 0 = n2.
+#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H) ]
+/2 width=1 by or_introl, or_intror/
qed-.
-lemma lveq_fwd_pair_sn: ∀I,K1,L2,V,n1,n2. K1.ⓑ{I}V ≋ⓧ*[n1, n2] L2 → 0 = n1.
-/2 width=8 by lveq_fwd_pair_sn_aux/ qed-.
+lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1.
+#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
+]
+qed-.
-lemma lveq_fwd_pair_dx: ∀I,L1,K2,V,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I}V → 0 = n2.
+lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 0 = n2.
/3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.
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