X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fsyntax%2Flveq.ma;h=776affc1c95b4f3fda5e8dddb22ef4bb39424c59;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=855978783c87ffefc69f70fc69d6fe0de0e39d8b;hpb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
index 855978783..776affc1c 100644
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
@@ -12,6 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_3_4.ma".
+include "ground_2/xoa/ex_4_1.ma".
 include "static_2/notation/relations/voidstareq_4.ma".
 include "static_2/syntax/lenv.ma".
 
@@ -20,7 +22,7 @@ include "static_2/syntax/lenv.ma".
 inductive lveq: bi_relation nat lenv ≝
 | lveq_atom   : lveq 0 (⋆) 0 (⋆)
 | lveq_bind   : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 →
-                lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓘ{I2})
+                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 →
@@ -32,7 +34,7 @@ interpretation "equivalence up to exclusion binders (local environment)"
 
 (* Basic properties *********************************************************)
 
-lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L.
+lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L.
 #L elim L -L /2 width=1 by lveq_atom, lveq_bind/
 qed.
 
@@ -43,10 +45,10 @@ qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+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.
+                            | ∃∃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/
@@ -54,14 +56,14 @@ fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
 ]
 qed-.
 
-lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
+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.
+                      | ∃∃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 →
+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.
+                           ∃∃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
@@ -69,18 +71,18 @@ fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
 ]
 qed-.
 
-lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, n2] K2 →
-                        ∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2.
+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.
+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 →
+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
 [1: #m1 #m2 #H1 #H2 destruct
@@ -89,17 +91,17 @@ fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
 ]
 qed-.
 
-lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥.
+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_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2.
+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.
+
+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
@@ -108,8 +110,8 @@ lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 &
 ]
 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.
+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
@@ -121,16 +123,16 @@ lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
 ]
 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.
+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.
+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
@@ -142,14 +144,14 @@ lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n
 ]
 qed-.
 
-lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, n2] L2 →
-                             ∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2.
+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.
+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
@@ -158,19 +160,19 @@ qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+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: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1.
+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: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 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-.