more additions and corrections for the article

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / static / lsubf.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma b/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
index 7056a8ec6fca39b8613a2db0ed5d4722214647db..e2ac985dc19f77c1e57615eb95eb1e71acbfe4c0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
@@ -24,9 +24,9 @@ inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
               lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
 | lsubf_bind: ∀f1,f2,I,L1,L2. lsubf L1 f1 L2 f2 →
               lsubf (L1.ⓘ{I}) (↑f1) (L2.ⓘ{I}) (↑f2)
-| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
               lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
-| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
               lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
 .
 
@@ -36,8 +36,9 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L1 = ⋆ →
-                          f1 ≡ f2 ∧ L2 = ⋆.
+fact lsubf_inv_atom1_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L1 = ⋆ →
+     ∧∧ f1 ≡ f2 & L2 = ⋆.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ /2 width=1 by conj/
 | #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
@@ -47,12 +48,13 @@ fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L
 ]
 qed-.
 
-lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
+lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → ∧∧ f1 ≡ f2 & L2 = ⋆.
 /2 width=3 by lsubf_inv_atom1_aux/ qed-.
 
-fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                          ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
-                          ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+fact lsubf_inv_push1_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+     ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
+     ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
@@ -63,19 +65,20 @@ fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
 ]
 qed-.
 
-lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                       ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+lemma lsubf_inv_push1:
+      ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+      ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
 /2 width=6 by lsubf_inv_push1_aux/ qed-.
 
-fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                          ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
-                          ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
-                           | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & 
-                                               K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
-                                               I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
-                           | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & 
-                                             K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
-                                             L2 = K2.ⓤ{J}.
+fact lsubf_inv_pair1_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+     ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
+     ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+      | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & 
+          K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+          I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
+      | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & 
+          K1 ⊢ 𝐅+⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 & L2 = K2.ⓤ{J}.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
@@ -88,19 +91,20 @@ fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
 ]
 qed-.
 
-lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X,↑g1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                       ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
-                        | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & 
-                                            K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
-                                            I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
-                        | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & 
-                                          K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
-                                          L2 = K2.ⓤ{J}.
+lemma lsubf_inv_pair1:
+      ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X,↑g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+      ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+       | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & 
+           K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+           I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
+       | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & 
+           K1 ⊢ 𝐅+⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 & L2 = K2.ⓤ{J}.
 /2 width=5 by lsubf_inv_pair1_aux/ qed-.
 
-fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                          ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
-                          ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+fact lsubf_inv_unit1_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+     ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
+     ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
@@ -111,12 +115,14 @@ fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
 ]
 qed-.
 
-lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I},↑g1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                       ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+lemma lsubf_inv_unit1:
+      ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I},↑g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+      ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
 /2 width=5 by lsubf_inv_unit1_aux/ qed-.
 
-fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L2 = ⋆ →
-                          f1 ≡ f2 ∧ L1 = ⋆.
+fact lsubf_inv_atom2_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L2 = ⋆ →
+     ∧∧ f1 ≡ f2 & L1 = ⋆.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ /2 width=1 by conj/
 | #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
@@ -126,12 +132,13 @@ fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L
 ]
 qed-.
 
-lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1,f1⦄ ⫃𝐅* ⦃⋆,f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
+lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1,f1⦄ ⫃𝐅+ ⦃⋆,f2⦄ → ∧∧f1 ≡ f2 & L1 = ⋆.
 /2 width=3 by lsubf_inv_atom2_aux/ qed-.
 
-fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                          ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
-                          ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+fact lsubf_inv_push2_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+     ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
+     ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
@@ -142,16 +149,18 @@ fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
 ]
 qed-.
 
-lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1,f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2},⫯g2⦄ →
-                       ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+lemma lsubf_inv_push2:
+      ∀f1,g2,I2,L1,K2. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓘ{I2},⫯g2⦄ →
+      ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
 /2 width=6 by lsubf_inv_push2_aux/ qed-.
 
-fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                          ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
-                          ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
-                           | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
-                                             K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
-                                             I = Abst & L1 = K1.ⓓⓝW.V.
+fact lsubf_inv_pair2_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+     ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
+     ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+      | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+          K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+          I = Abst & L1 = K1.ⓓⓝW.V.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
@@ -163,19 +172,20 @@ fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
 ]
 qed-.
 
-lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1,f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W,↑g2⦄ →
-                       ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
-                        | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ &
-                                          K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
-                                          I = Abst & L1 = K1.ⓓⓝW.V.
+lemma lsubf_inv_pair2:
+      ∀f1,g2,I,L1,K2,W. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓑ{I}W,↑g2⦄ →
+      ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+       | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+           K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+           I = Abst & L1 = K1.ⓓⓝW.V.
 /2 width=5 by lsubf_inv_pair2_aux/ qed-.
 
-fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                          ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
-                          ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
-                           | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & 
-                                               K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
-                                               L1 = K1.ⓑ{J}V.
+fact lsubf_inv_unit2_aux:
+     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+     ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
+     ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+      | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & 
+          K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 & L1 = K1.ⓑ{J}V.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
 | #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
@@ -187,27 +197,29 @@ fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
 ]
 qed-.
 
-lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1,f1⦄ ⫃𝐅* ⦃K2.ⓤ{I},↑g2⦄ →
-                       ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
-                        | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & 
-                                            K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
-                                            L1 = K1.ⓑ{J}V.
+lemma lsubf_inv_unit2:
+      ∀f1,g2,I,L1,K2. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓤ{I},↑g2⦄ →
+      ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+       | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & 
+           K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 & L1 = K1.ⓑ{J}V.
 /2 width=5 by lsubf_inv_unit2_aux/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆,f1⦄ ⫃𝐅* ⦃⋆,f2⦄ → f1 ≡ f2.
+lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆,f1⦄ ⫃𝐅+ ⦃⋆,f2⦄ → f1 ≡ f2.
 #f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
 qed-.
 
-lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2},f2⦄ →
-                         ∃∃g2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ⫯g2.
+lemma lsubf_inv_push_sn:
+      ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅+ ⦃K2.ⓘ{I2},f2⦄ →
+      ∃∃g2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2.
 #g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
 #g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
 qed-.
 
-lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I},↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I},f2⦄ →
-                         ∃∃g2. ⦃K1,g1⦄ ⫃𝐅* ⦃K2,g2⦄ & f2 = ↑g2.
+lemma lsubf_inv_bind_sn:
+      ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I},↑g1⦄ ⫃𝐅+ ⦃K2.ⓘ{I},f2⦄ →
+      ∃∃g2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2.
 #g1 #f2 * #I [2: #X ] #K1 #K2 #H
 [ elim (lsubf_inv_pair1 … H) -H *
   [ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
@@ -219,8 +231,9 @@ lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I},↑g1⦄ ⫃𝐅* ⦃K2.
 ]
 qed-.
 
-lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V,↑g1⦄ ⫃𝐅* ⦃K2.ⓛW,f2⦄ →
-                         ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+lemma lsubf_inv_beta_sn:
+      ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V,↑g1⦄ ⫃𝐅+ ⦃K2.ⓛW,f2⦄ →
+      ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
 #g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
 [ #z2 #Y2 #_ #_ #H destruct
 | #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
@@ -229,8 +242,9 @@ lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V,↑g1⦄ ⫃𝐅* 
 ]
 qed-.
 
-lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J},f2⦄ →
-                         ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅* ⦃K2,g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+lemma lsubf_inv_unit_sn:
+      ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅+ ⦃K2.ⓤ{J},f2⦄ →
+      ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
 #g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
 [ #z2 #Y2 #_ #_ #H destruct
 | #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
@@ -239,7 +253,7 @@ lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅* 
 ]
 qed-.
 
-lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄ → f1 ≡ f2.
+lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅+ ⦃L,f2⦄ → f1 ≡ f2.
 #L elim L -L /2 width=1 by lsubf_inv_atom/
 #L #I #IH #f1 #f2 #H12
 elim (pn_split f1) * #g1 #H destruct
@@ -249,15 +263,15 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lsubf_fwd_bind_tl: ∀f1,f2,I,L1,L2.
-                         ⦃L1.ⓘ{I},f1⦄ ⫃𝐅* ⦃L2.ⓘ{I},f2⦄ → ⦃L1,⫱f1⦄ ⫃𝐅* ⦃L2,⫱f2⦄.
+lemma lsubf_fwd_bind_tl:
+      ∀f1,f2,I,L1,L2. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅+ ⦃L2.ⓘ{I},f2⦄ → ⦃L1,⫱f1⦄ ⫃𝐅+ ⦃L2,⫱f2⦄.
 #f1 #f2 #I #L1 #L2 #H
 elim (pn_split f1) * #g1 #H0 destruct
 [ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
 #g2 #H12 #H destruct //
 qed-.
 
-lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
+lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
 [ /2 width=3 by isid_eq_repl_fwd/
 | /4 width=3 by isid_inv_push, isid_push/
@@ -267,7 +281,7 @@ lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 
 ]
 qed-.
 
-lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
+lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
 [ /2 width=3 by isid_eq_repl_back/
 | /4 width=3 by isid_inv_push, isid_push/
@@ -277,22 +291,22 @@ lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 
 ]
 qed-.
 
-lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → f2 ⊆ f1.
+lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → f2 ⊆ f1.
 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
 /3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
 qed-.
 
 (* Basic properties *********************************************************)
 
-axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
+axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
 
-lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
+lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
 #f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
 qed-.
 
-axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
+axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
 
-lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄).
+lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
 #f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
 qed-.
 
@@ -302,19 +316,21 @@ lemma lsubf_refl: bi_reflexive … lsubf.
 /2 width=1 by lsubf_push, lsubf_bind/
 qed.
 
-lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄.
+lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L,f1⦄ ⫃𝐅+ ⦃L,f2⦄.
 /2 width=3 by lsubf_eq_repl_back2/ qed.
 
-lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1,g1⦄ ⫃𝐅* ⦃L2,⫱f2⦄ →
-                        ∃∃f1. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅* ⦃L2.ⓘ{I},f2⦄ & g1 = ⫱f1.
+lemma lsubf_bind_tl_dx:
+      ∀g1,f2,I,L1,L2. ⦃L1,g1⦄ ⫃𝐅+ ⦃L2,⫱f2⦄ →
+      ∃∃f1. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅+ ⦃L2.ⓘ{I},f2⦄ & g1 = ⫱f1.
 #g1 #f2 #I #L1 #L2 #H
 elim (pn_split f2) * #g2 #H2 destruct
 @ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
 qed-.
 
-lemma lsubf_beta_tl_dx: ∀f,f0,g1,L1,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
-                        ∀f2,L2,W. ⦃L1,f0⦄ ⫃𝐅* ⦃L2,⫱f2⦄ →
-                        ∃∃f1. ⦃L1.ⓓⓝW.V,f1⦄ ⫃𝐅* ⦃L2.ⓛW,f2⦄ & ⫱f1 ⊆ g1.
+lemma lsubf_beta_tl_dx:
+      ∀f,f0,g1,L1,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
+      ∀f2,L2,W. ⦃L1,f0⦄ ⫃𝐅+ ⦃L2,⫱f2⦄ →
+      ∃∃f1. ⦃L1.ⓓⓝW.V,f1⦄ ⫃𝐅+ ⦃L2.ⓛW,f2⦄ & ⫱f1 ⊆ g1.
 #f #f0 #g1 #L1 #V #Hf #Hg1 #f2
 elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
 [ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
@@ -323,9 +339,10 @@ elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
 qed-.
 
 (* Note: this might be moved *)
-lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ →
-                        ∀f2l,f2r. f2l⋓f2r ≘ f2 →
-                        ∃∃f1l,f1r. ⦃L1,f1l⦄ ⫃𝐅* ⦃L2,f2l⦄ & ⦃L1,f1r⦄ ⫃𝐅* ⦃L2,f2r⦄ & f1l⋓f1r ≘ f1.
+lemma lsubf_inv_sor_dx:
+      ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+      ∀f2l,f2r. f2l⋓f2r ≘ f2 →
+      ∃∃f1l,f1r. ⦃L1,f1l⦄ ⫃𝐅+ ⦃L2,f2l⦄ & ⦃L1,f1r⦄ ⫃𝐅+ ⦃L2,f2r⦄ & f1l⋓f1r ≘ f1.
 #f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
 [ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
 | #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H