work in progress with voids and lveq (was: the most recent voids)

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / syntax / lveq_lveq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/lveq_lveq.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/lveq_lveq.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/syntax/lveq_length.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
+
+(* 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
+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/
+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 _/
+]
+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/
+]
+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-.
+*)
+*)
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