renaming in basic_2

[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
index 3f64c33b02d7ee4b753a04bcea7a060b5770dce5..40a5fb5c86152316956817fe42f548133bdcebf6 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/syntax/lveq_lveq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/lveq_lveq.ma
@@ -16,131 +16,38 @@ include "basic_2/syntax/lveq_length.ma".
 
 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
 
-(* Main forward lemmas ******************************************************)
-
-theorem lveq_fwd_inj_succ_zero: ∀L1,L2,n1. L1 ≋ⓧ*[⫯n1, 0] L2 →
-                                ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 → ∃x1. ⫯x1 = m1.
-#L1 #L2 #n1 #Hn #m1 #m2 #Hm
-lapply (lveq_fwd_length … Hn) -Hn <plus_n_O #Hn
-lapply (lveq_fwd_length … Hm) -Hm >Hn >associative_plus -Hn #Hm
-lapply (injective_plus_r … Hm) -Hm
-<plus_S1 /2 width=2 by ex_intro/
-qed-.
-
-theorem lveq_fwd_inj_zero_succ: ∀L1,L2,n2. L1 ≋ⓧ*[0, ⫯n2] L2 →
-                                ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 → ∃x2. ⫯x2 = m2.
-/4 width=6 by lveq_fwd_inj_succ_zero, lveq_sym/ qed-. (* auto: 2x lveq_sym *)
-
-theorem lveq_fwd_inj_succ_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[⫯n1, n2] L2 →
-                              ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
-                              ∨∨ ∃x. ⫯x = n2 | ∃x. ⫯x = m1.
-#L1 #L2 #n1 * [2: #n2 ] /3 width=2 by ex_intro, or_introl/
-#Hn #m1 #m2 #Hm @or_intror @lveq_fwd_inj_succ_zero /width=6 by/ (**) (* auto fails *)
-qed-.
-
-theorem lveq_fwd_inj_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ⫯n2] L2 →
-                              ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
-                              ∨∨ ∃x. ⫯x = n1 | ∃x. ⫯x = m2.
-/4 width=6 by lveq_fwd_inj_succ_sn, lveq_sym/ qed-. (* auto: 2x lveq_sym *) 
-
 (* 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_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 * [2: #L1 #I1 ] * [2,4: #L2 #I2 ]
-[ cases I1 -I1 [ * | #I1 #V1 ] cases I2 -I2 [1,3: * |*: #I2 #V2 ] ]
-#Hx #n1 #n2 #Hn #m1 #m2 #Hm destruct
-[ elim (lveq_fwd_void_void … Hn) * #x #H destruct
-  elim (lveq_fwd_void_void … Hm) * #y #H destruct
-  [ lapply (lveq_inv_void_succ_sn … Hn) -Hn #Hn
-    lapply (lveq_inv_void_succ_sn … Hm) -Hm #Hm
-    elim (IH … Hn … Hm) -IH -Hn -Hm // #H1 #H2 destruct
-    /2 width=1 by conj/
-  | elim (lveq_fwd_inj_succ_sn … Hn … Hm) * #z #H destruct
-    [ lapply (lveq_inv_void_succ_dx … Hn) -Hn #Hn
-      lapply (lveq_inv_void_succ_dx … Hm) -Hm #Hm
-      elim (IH … Hn … Hm) -IH -Hn -Hm [2: normalize // ] #H1 #H2 destruct (**) (* avoid normalize *)
-      /2 width=1 by conj/
-    | lapply (lveq_inv_void_succ_sn … Hn) -Hn #Hn
-      lapply (lveq_inv_void_succ_sn … Hm) -Hm #Hm
-      elim (IH … Hn … Hm) -IH -Hn -Hm // #H1 #H2 destruct
-      /2 width=1 by conj/
-    ]
-  | elim (lveq_fwd_inj_succ_dx … Hn … Hm) * #z #H destruct
-    [ lapply (lveq_inv_void_succ_sn … Hn) -Hn #Hn
-      lapply (lveq_inv_void_succ_sn … Hm) -Hm #Hm
-      elim (IH … Hn … Hm) -IH -Hn -Hm // #H1 #H2 destruct
-      /2 width=1 by conj/
-    | lapply (lveq_inv_void_succ_dx … Hn) -Hn #Hn
-      lapply (lveq_inv_void_succ_dx … Hm) -Hm #Hm
-      elim (IH … Hn … Hm) -IH -Hn -Hm [2: normalize // ] #H1 #H2 destruct (**) (* avoid normalize *)
-      /2 width=1 by conj/
-    ]
-  | lapply (lveq_inv_void_succ_dx … Hn) -Hn #Hn
-    lapply (lveq_inv_void_succ_dx … Hm) -Hm #Hm
-    elim (IH … Hn … Hm) -IH -Hn -Hm [2: normalize // ] #H1 #H2 destruct (**) (* avoid normalize *)
-    /2 width=1 by conj/
-  ]
-| lapply (lveq_fwd_abst_bind_length_le … Hn) #HL
-  elim (le_to_or_lt_eq … HL) -HL #HL
-  [ elim (lveq_inv_void_dx_length … Hn) -Hn // #x1 #Hn #H #_ destruct
-    elim (lveq_inv_void_dx_length … Hm) -Hm // #y1 #Hm #H #_ destruct
-    elim (IH … Hn … Hm) -IH -Hn -Hm -HL [2: normalize // ] #H1 #H2 destruct (**) (* avoid normalize *)
-    /2 width=1 by conj/
-  | elim (lveq_eq_ex … HL) -HL #x #HL
-    elim (lveq_inv_pair_sn … HL … Hn) -Hn #H1 #H2 destruct
-    elim (lveq_inv_pair_sn … HL … Hm) -Hm #H1 #H2 destruct
-    /2 width=1 by conj/
-  ]
-| lapply (lveq_fwd_bind_abst_length_le … Hn) #HL
-  elim (le_to_or_lt_eq … HL) -HL #HL
-  [ elim (lveq_inv_void_sn_length … Hn) -Hn // #x1 #Hn #H #_ destruct
-    elim (lveq_inv_void_sn_length … Hm) -Hm // #y1 #Hm #H #_ destruct
-    elim (IH … Hn … Hm) -IH -Hn -Hm -HL // #H1 #H2 destruct
-    /2 width=1 by conj/
-  | lapply (sym_eq ??? HL) -HL #HL
-    elim (lveq_eq_ex … HL) -HL #x #HL
-    elim (lveq_inv_pair_dx … HL … Hn) -Hn #H1 #H2 destruct
-    elim (lveq_inv_pair_dx … HL … Hm) -Hm #H1 #H2 destruct
-    /2 width=1 by conj/
-  ]
-| elim (lveq_inv_pair_pair… Hn) -Hn #x #_ #H1 #H2 destruct
-  elim (lveq_inv_pair_pair… Hm) -Hm #y #_ #H1 #H2 destruct
-  /2 width=1 by conj/
-| elim (lveq_inv_atom_bind … Hn) -Hn #x #Hn #H1 #H2 destruct
-  elim (lveq_inv_atom_bind … Hm) -Hm #y #Hm #H1 #H2 destruct
-  elim (IH … Hn … Hm) -IH -Hn -Hm /2 width=1 by conj/
-| elim (lveq_inv_bind_atom … Hn) -Hn #x #Hn #H1 #H2 destruct
-  elim (lveq_inv_bind_atom … Hm) -Hm #y #Hm #H1 #H2 destruct
-  elim (IH … Hn … Hm) -IH -Hn -Hm /2 width=1 by conj/
-| elim (lveq_inv_atom_atom … Hn) -Hn #H1 #H2 destruct
-  elim (lveq_inv_atom_atom … Hm) -Hm #H1 #H2 destruct
-  /2 width=1 by conj/
-]
+#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-.
 
-theorem lveq_inj_void_sn: ∀K1,K2,n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
-                          ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
-                          ∧∧ ⫯n1 = m1 & n2 = m2.
-/3 width=4 by lveq_inj, lveq_void_sn/ qed-.
+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-.
 
-theorem lveq_inj_void_dx: ∀K1,K2,n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
-                          ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ →
-                          ∧∧ n1 = m1 & ⫯n2 = m2.
-/3 width=4 by lveq_inj, lveq_void_dx/ qed-.
+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 *)