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[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / lveq.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
index 855978783c87ffefc69f70fc69d6fe0de0e39d8b..e6d2fcbfc37a30a8714ffa23f1451faf90f02da6 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
@@ -32,7 +32,7 @@ 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 +43,10 @@ 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 +54,14 @@ 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 +69,18 @@ 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 +89,17 @@ 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 +108,8 @@ 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 +121,16 @@ 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 +142,14 @@ 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 +158,19 @@ 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-.