(* Main inversion lemmas ****************************************************)
-theorem lveq_inv_pair_sn: ∀K1,K2,n. K1 ≋ⓧ*[n, n] K2 →
- ∀I1,I2,V,m1,m2. K1.ⓑ{I1}V ≋ⓧ*[m1, m2] K2.ⓘ{I2} →
- ∧∧ 0 = m1 & 0 = m2.
-#K1 #K2 #n #HK #I1 #I2 #V #m1 #m2 #H
+theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0, 0] K2 →
+ ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓘ{I2} →
+ ∧∧ 0 = m1 & 0 = m2.
+#K1 #K2 #HK #I1 #I2 #m1 #m2 #H
lapply (lveq_fwd_length_eq … HK) -HK #HK
-lapply (lveq_fwd_pair_sn … H) #H0 destruct
-<(lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/
+elim (lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/
qed-.
-theorem lveq_inv_pair_dx: ∀K1,K2,n. K1 ≋ⓧ*[n, n] K2 →
- ∀I1,I2,V,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓑ{I2}V →
- ∧∧ 0 = m1 & 0 = m2.
-/4 width=8 by lveq_inv_pair_sn, lveq_sym, commutative_and/ qed-.
-(*
-theorem lveq_inv_void_sn: ∀K1,K2,n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
- 0 < m1.
-*)
-(*
theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
∧∧ n1 = m1 & n2 = m2.
-#L1 #L2 @(f2_ind ?? length2 ?? L1 L2) -L1 -L2
-#x #IH #L1 #L2 #Hx #n1 #n2 #H
-generalize in match Hx; -Hx
-cases H -L1 -L2 -n1 -n2
-/2 width=8 by lveq_inv_pair_dx, lveq_inv_pair_sn, lveq_inv_atom/
-#K1 #K2 #n1 #n2 #HK #Hx #m1 #m2 #H destruct
-
-
-
-[ #_ #m1 #m2 #HL -x /2 width=1 by lveq_inv_atom/
-| #I1 #I2 #K1 #K2 #V1 #n #HK #_ #m1 #m2 #H -x
-
-
-
-theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
- ∧∧ n1 = m1 & n2 = m2.
-#L1 #L2 #n1 #n2 #H @(lveq_ind_voids … H) -H -L1 -L2 -n1 -n2
-[ #n1 #n2 #m1 #m2 #H elim (lveq_inv_voids … H) -H *
- [ /3 width=1 by voids_inj, conj/ ]
- #J1 #J2 #K1 #K2 #W #m #_ [ #H #_ | #_ #H ]
- elim (voids_inv_pair_sn … H) -H #H #_
- elim (voids_atom_inv … H) -H #H #_ destruct
-]
-#I1 #I2 #L1 #L2 #V #n1 #n2 #n #HL #IH #m1 #m2 #H
-elim (lveq_inv_voids … H) -H *
-[1,4: [ #H #_ | #_ #H ]
- elim (voids_inv_atom_sn … H) -H #H #_
- elim (voids_pair_inv … H) -H #H #_ destruct
-]
-#J1 #J2 #K1 #K2 #W #m #HK [1,3: #H1 #H2 |*: #H2 #H1 ]
-elim (voids_inv_pair_sn … H1) -H1 #H #Hnm
-[1,4: -IH -Hnm elim (voids_pair_inv … H) -H #H1 #H2 destruct
-|2,3: elim (voids_inv_pair_dx … H2) -H2 #H2 #_
-
- elim (IH … HK)
-
-
-(*
-/3 width=3 by lveq_inv_atom, lveq_inv_voids/
-|
- lapply (lveq_inv_voids … H) -H #H
- elim (lveq_inv_pair_sn … H) -H * /2 width=1 by conj/
- #Y2 #y2 #HY2 #H1 #H2 #H3 destruct
-*)
-
-(*
-fact lveq_inv_pair_bind_aux: ∀L1,L2,n1,n2. L1 ≋ ⓧ*[n1, n2] L2 →
- ∀I1,I2,K1,K2,V1. K1.ⓑ{I1}V1 = L1 → K2.ⓘ{I2} = L2 →
- ∨∨ ∃∃m. K1 ≋ ⓧ*[m, m] K2 & 0 = n1 & 0 = n2
- | ∃∃m1,m2. K1 ≋ ⓧ*[m1, m2] K2 &
- BUnit Void = I2 & ⫯m2 = n2.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2
-[ #J1 #J2 #L1 #L2 #V1 #H1 #H2 destruct
-|2,3: #I1 #I2 #K1 #K2 #V #n #HK #_ #J1 #J2 #L1 #L2 #V1 #H1 #H2 destruct /3 width=2 by or_introl, ex3_intro/
-|4,5: #K1 #K2 #n1 #n2 #HK #IH #J1 #J2 #L1 #L2 #V1 #H1 #H2 destruct
- /3 width=4 by _/
-]
+#L1 #L2 #n1 #n2 #Hn #m1 #m2 #Hm
+elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
+elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct
+/2 width=1 by conj/
qed-.
-lemma voids_inv_pair_bind: ∀I1,I2,K1,K2,V1,n1,n2. ⓧ*[n1]K1.ⓑ{I1}V1 ≋ ⓧ*[n2]K2.ⓘ{I2} →
- ∨∨ ∃∃n. ⓧ*[n]K1 ≋ ⓧ*[n]K2 & 0 = n1 & 0 = n2
- | ∃∃m2. ⓧ*[n1]K1.ⓑ{I1}V1 ≋ ⓧ*[m2]K2 &
- BUnit Void = I2 & ⫯m2 = n2.
-/2 width=5 by voids_inv_pair_bind_aux/ qed-.
-
-fact voids_inv_bind_pair_aux: ∀L1,L2,n1,n2. ⓧ*[n1]L1 ≋ ⓧ*[n2]L2 →
- ∀I1,I2,K1,K2,V2. K1.ⓘ{I1} = L1 → K2.ⓑ{I2}V2 = L2 →
- ∨∨ ∃∃n. ⓧ*[n]K1 ≋ ⓧ*[n]K2 & 0 = n1 & 0 = n2
- | ∃∃m1. ⓧ*[m1]K1 ≋ ⓧ*[n2]K2.ⓑ{I2}V2 &
- BUnit Void = I1 & ⫯m1 = n1.
-#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[ #J1 #J2 #L1 #L2 #V1 #H1 #H2 destruct
-|2,3: #I1 #I2 #K1 #K2 #V #n #HK #J1 #J2 #L1 #L2 #V2 #H1 #H2 destruct /3 width=2 by or_introl, ex3_intro/
-|4,5: #K1 #K2 #n1 #n2 #HK #J1 #J2 #L1 #L2 #V2 #H1 #H2 destruct /3 width=3 by or_intror, ex3_intro/
-]
+theorem lveq_inj_void_sn_ge: ∀K1,K2. |K2| ≤ |K1| →
+ ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
+ ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
+ ∧∧ ↑n1 = m1 & 0 = m2 & 0 = n2.
+#L1 #L2 #HL #n1 #n2 #Hn #m1 #m2 #Hm
+elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
+elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct
+>length_bind >eq_minus_S_pred >(eq_minus_O … HL)
+/3 width=4 by plus_minus, and3_intro/
qed-.
-lemma voids_inv_bind_pair: ∀I1,I2,K1,K2,V2,n1,n2. ⓧ*[n1]K1.ⓘ{I1} ≋ ⓧ*[n2]K2.ⓑ{I2}V2 →
- ∨∨ ∃∃n. ⓧ*[n]K1 ≋ ⓧ*[n]K2 & 0 = n1 & 0 = n2
- | ∃∃m1. ⓧ*[m1]K1 ≋ ⓧ*[n2]K2.ⓑ{I2}V2 &
- BUnit Void = I1 & ⫯m1 = n1.
-/2 width=5 by voids_inv_bind_pair_aux/ qed-.
-*)
-*)
\ No newline at end of file
+theorem lveq_inj_void_dx_le: ∀K1,K2. |K1| ≤ |K2| →
+ ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
+ ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ →
+ ∧∧ ↑n2 = m2 & 0 = m1 & 0 = n1.
+/3 width=5 by lveq_inj_void_sn_ge, lveq_sym/ qed-. (* auto: 2x lveq_sym *)
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