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[helm.git] / matita / matita / contribs / lambdadelta / static_2 / i_static / rexs.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma
index 8beb5bc3f7b93fee884647bc2297aa7b43574a8a..a53ee2c823636520601d31342a6210a725efc554 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma
@@ -25,50 +25,50 @@ interpretation "iterated extension on referred entries (local environment)"
 
 (* Basic properties *********************************************************)
 
-lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L →
-                    ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R,T] L →
+                    ∀L2. L ⪤[R,T] L2 → L1 ⪤*[R,T] L2.
 #R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
 qed-.
 
-lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L →
-                    ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R,T] L →
+                    ∀L2. L ⪤*[R,T] L2 → L1 ⪤*[R,T] L2.
 #R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
 qed-.
 
-lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
+lemma rexs_atom: ∀R,I. ⋆ ⪤*[R,⓪{I}] ⋆.
 /2 width=1 by inj/ qed.
 
 lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s.
-                 L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
+                 L1 ⪤*[R,⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R,⋆s] L2.ⓑ{I}V2.
 #R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
 /3 width=4 by rex_sort, rexs_step_dx, inj/
 qed.
 
 lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) →
-                 ∀I,L1,L2,V. L1 ⪤*[R, V] L2 →
-                 L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V.
+                 ∀I,L1,L2,V. L1 ⪤*[R,V] L2 →
+                 L1.ⓑ{I}V ⪤*[R,#0] L2.ⓑ{I}V.
 #R #HR #I #L1 #L2 #V #H elim H -L2
 /3 width=5 by rex_pair, rexs_step_dx, inj/
 qed.
 
-lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
-                 L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
+lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 →
+                 L1.ⓤ{I} ⪤*[R,#0] L2.ⓤ{I}.
 /3 width=3 by rex_unit, inj/ qed.
 
 lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i.
-                 L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2.
+                 L1 ⪤*[R,#i] L2 → L1.ⓑ{I}V1 ⪤*[R,#↑i] L2.ⓑ{I}V2.
 #R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
 /3 width=4 by rex_lref, rexs_step_dx, inj/
 qed.
 
 lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l.
-                 L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
+                 L1 ⪤*[R,§l] L2 → L1.ⓑ{I}V1 ⪤*[R,§l] L2.ⓑ{I}V2.
 #R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
 /3 width=4 by rex_gref, rexs_step_dx, inj/
 qed.
 
 lemma rexs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
-               ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
+               ∀L1,L2,T. L1 ⪤*[R1,T] L2 → L1 ⪤*[R2,T] L2.
 #R1 #R2 #HR #L1 #L2 #T #H elim H -L2
 /4 width=5 by rex_co, rexs_step_dx, inj/
 qed-.
@@ -76,19 +76,19 @@ qed-.
 (* Basic inversion lemmas ***************************************************)
 
 (* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
-lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆.
+lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R,⓪{I}] Y2 → Y2 = ⋆.
 #R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/
 qed-.
 
 (* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
-lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R,⓪{I}] ⋆ → Y1 = ⋆.
 #R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
 /3 width=3 by inj, rex_inv_atom_dx/
 qed-.
 
-lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
+lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R,⋆s] Y2 →
                      ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
-                      | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
+                      | ∃∃I1,I2,L1,L2. L1 ⪤*[R,⋆s] L2 &
                                        Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
 #R #Y1 #Y2 #s #H elim H -Y2
 [ #Y2 #H elim (rex_inv_sort … H) -H *
@@ -104,9 +104,9 @@ lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
 ] 
 qed-.
 
-lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
+lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R,§l] Y2 →
                      ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
-                      | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
+                      | ∃∃I1,I2,L1,L2. L1 ⪤*[R,§l] L2 &
                                        Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
 #R #Y1 #Y2 #l #H elim H -Y2
 [ #Y2 #H elim (rex_inv_gref … H) -H *
@@ -123,16 +123,16 @@ lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
 qed-.
 
 lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
-                     ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
-                     ∧∧ L1 ⪤*[R, V] L2 & L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+                     ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+                     ∧∧ L1 ⪤*[R,V] L2 & L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V.
 #R #HR #p #I #L1 #L2 #V #T #H elim H -L2
 [ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/
 | #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/
 ]
 qed-.
 
-lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
-                     ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2.
+lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 →
+                     ∧∧ L1 ⪤*[R,V] L2 & L1 ⪤*[R,T] L2.
 #R #I #L1 #L2 #V #T #H elim H -L2
 [ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/
 | #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/
@@ -141,32 +141,32 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 →
-                             ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R,⋆s] Y2 →
+                             ∃∃I2,L2. L1 ⪤*[R,⋆s] L2 & Y2 = L2.ⓘ{I2}.
 #R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H *
 [ #H destruct
 | #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 ]
 qed-.
 
-lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} →
-                             ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R,⋆s] L2.ⓘ{I2} →
+                             ∃∃I1,L1. L1 ⪤*[R,⋆s] L2 & Y1 = L1.ⓘ{I1}.
 #R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H *
 [ #_ #H destruct
 | #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 ]
 qed-.
 
-lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 →
-                             ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}.
+lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R,§l] Y2 →
+                             ∃∃I2,L2. L1 ⪤*[R,§l] L2 & Y2 = L2.ⓘ{I2}.
 #R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H *
 [ #H destruct
 | #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 ]
 qed-.
 
-lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} →
-                             ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}.
+lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R,§l] L2.ⓘ{I2} →
+                             ∃∃I1,L1. L1 ⪤*[R,§l] L2 & Y1 = L1.ⓘ{I1}.
 #R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H *
 [ #_ #H destruct
 | #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
@@ -175,18 +175,18 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
+lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R,②{I}V.T] L2 → L1 ⪤*[R,V] L2.
 #R #I #L1 #L2 #V #T #H elim H -L2
 /3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/
 qed-.
 
 lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
-                        ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
-                        L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+                        ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+                        L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V.
 #R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H //
 qed-.
 
-lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 → L1 ⪤*[R,T] L2.
 #R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H //
 qed-.