update in ground_2, static_2, basic_2, apps_2, alpha_1

[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..2c3a1bf375f906234d686345b560fb5fe7e0c51e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
@@ -12,6 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_3_3.ma".
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/xoa/ex_5_5.ma".
+include "ground_2/xoa/ex_5_6.ma".
+include "ground_2/xoa/ex_6_5.ma".
+include "ground_2/xoa/ex_7_6.ma".
 include "static_2/notation/relations/lrsubeqf_4.ma".
 include "ground_2/relocation/nstream_sor.ma".
 include "static_2/static/frees.ma".
@@ -21,13 +27,13 @@ include "static_2/static/frees.ma".
 inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
 | lsubf_atom: ∀f1,f2. f1 ≡ f2 → lsubf (⋆) f1 (⋆) f2
 | lsubf_push: ∀f1,f2,I1,I2,L1,L2. lsubf L1 (f1) L2 (f2) →
-              lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
+              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 (L1.ⓘ[I]) (↑f1) (L2.ⓘ[I]) (↑f2)
+| 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 L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
+| 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)
 .
 
 interpretation
@@ -36,8 +42,9 @@ 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 +54,13 @@ fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L
 ]
 qed-.
 
-lemma lsubf_inv_atom1: â\88\80f1,f2,L2. â¦\83â\8b\86,f1â¦\84 â«\83ð\9d\90\85* â¦\83L2,f2â¦\84 â\86\92 f1 â\89¡ f2 â\88§ L2 = ⋆.
+lemma lsubf_inv_atom1: â\88\80f1,f2,L2. â\9dªâ\8b\86,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« â\86\92 â\88§â\88§ f1 â\89¡ 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 +71,20 @@ 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 +97,20 @@ 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 +121,14 @@ 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 +138,13 @@ fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → L
 ]
 qed-.
 
-lemma lsubf_inv_atom2: â\88\80f1,f2,L1. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85* â¦\83â\8b\86,f2â¦\84 â\86\92 f1 â\89¡ f2 â\88§ L1 = ⋆.
+lemma lsubf_inv_atom2: â\88\80f1,f2,L1. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªâ\8b\86,f2â\9d« â\86\92 â\88§â\88§f1 â\89¡ 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 +155,18 @@ 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 +178,20 @@ 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 +203,29 @@ 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: â\88\80f1,f2. â¦\83â\8b\86,f1â¦\84 â«\83ð\9d\90\85* â¦\83â\8b\86,f2â¦\84 → f1 ≡ f2.
+lemma lsubf_inv_atom: â\88\80f1,f2. â\9dªâ\8b\86,f1â\9d« â«\83ð\9d\90\85+ â\9dªâ\8b\86,f2â\9d« → 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 +237,9 @@ 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 +248,9 @@ 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 +259,7 @@ lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅* 
 ]
 qed-.
 
-lemma lsubf_inv_refl: â\88\80L,f1,f2. â¦\83L,f1â¦\84 â«\83ð\9d\90\85* â¦\83L,f2â¦\84 → f1 ≡ f2.
+lemma lsubf_inv_refl: â\88\80L,f1,f2. â\9dªL,f1â\9d« â«\83ð\9d\90\85+ â\9dªL,f2â\9d« → 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 +269,15 @@ 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: â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85* â¦\83L2,f2â¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+lemma lsubf_fwd_isid_dx: â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d«.
 #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 +287,7 @@ lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 
 ]
 qed-.
 
-lemma lsubf_fwd_isid_sn: â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85* â¦\83L2,f2â¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+lemma lsubf_fwd_isid_sn: â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d«.
 #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 +297,56 @@ lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅* ⦃L2,f2⦄ → 
 ]
 qed-.
 
-lemma lsubf_fwd_sle: â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85* â¦\83L2,f2â¦\84 → f2 ⊆ f1.
+lemma lsubf_fwd_sle: â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« → 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⦄).
+lemma lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ❪L1,f1❫ ⫃𝐅+ ❪L2,f2❫).
+#f2 #L1 #L2 #f #H elim H -f -f2 -L1 -L2
+[ #f1 #f2 #Hf12 #g1 #Hfg1
+  /3 width=3 by lsubf_atom, eq_canc_sn/
+| #f1 #f2 #I1 #I2 #K1 #K2 #_ #IH #g #H
+  elim (eq_inv_px … H) -H [|*: // ] #g1 #Hfg1 #H destruct
+  /3 width=1 by lsubf_push/
+| #f1 #f2 #I #K1 #K2 #_ #IH #g #H
+  elim (eq_inv_nx … H) -H [|*: // ] #g1 #Hfg1 #H destruct
+  /3 width=1 by lsubf_bind/
+| #f #f0 #f1 #f2 #K1 #L2 #W #V #Hf #Hf1 #_ #IH #g #H
+  elim (eq_inv_nx … H) -H [|*: // ] #g1 #Hfg1 #H destruct
+  /3 width=5 by lsubf_beta, sor_eq_repl_back3/
+| #f #f0 #f1 #f2 #I1 #I2 #K1 #K2 #V #Hf #Hf1 #_ #IH #g #H
+  elim (eq_inv_nx … H) -H [|*: // ] #g1 #Hfg1 #H destruct
+  /3 width=5 by lsubf_unit, sor_eq_repl_back3/
+]
+qed-.
 
-lemma lsubf_eq_repl_fwd1: â\88\80f2,L1,L2. eq_repl_fwd â\80¦ (λf1. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85* â¦\83L2,f2â¦\84).
+lemma lsubf_eq_repl_fwd1: â\88\80f2,L1,L2. eq_repl_fwd â\80¦ (λf1. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d«).
 #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⦄).
+lemma lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ❪L1,f1❫ ⫃𝐅+ ❪L2,f2❫).
+#f1 #L1 #L2 #f #H elim H -f1 -f -L1 -L2
+[ #f1 #f2 #Hf12 #g2 #Hfg2
+  /3 width=3 by lsubf_atom, eq_trans/
+| #f1 #f2 #I1 #I2 #K1 #K2 #_ #IH #g #H
+  elim (eq_inv_px … H) -H [|*: // ] #g2 #Hfg2 #H destruct
+  /3 width=1 by lsubf_push/
+| #f1 #f2 #I #K1 #K2 #_ #IH #g #H
+  elim (eq_inv_nx … H) -H [|*: // ] #g2 #Hfg2 #H destruct
+  /3 width=1 by lsubf_bind/
+| #f #f0 #f1 #f2 #K1 #L2 #W #V #Hf #Hf1 #_ #IH #g #H
+  elim (eq_inv_nx … H) -H [|*: // ] #g2 #Hfg2 #H destruct
+  /3 width=5 by lsubf_beta/
+| #f #f0 #f1 #f2 #I1 #I2 #K1 #K2 #V #Hf #Hf1 #_ #IH #g #H
+  elim (eq_inv_nx … H) -H [|*: // ] #g2 #Hfg2 #H destruct
+  /3 width=5 by lsubf_unit/
+]
+qed-.
 
-lemma lsubf_eq_repl_fwd2: â\88\80f1,L1,L2. eq_repl_fwd â\80¦ (λf2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85* â¦\83L2,f2â¦\84).
+lemma lsubf_eq_repl_fwd2: â\88\80f1,L1,L2. eq_repl_fwd â\80¦ (λf2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d«).
 #f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
 qed-.
 
@@ -302,30 +356,33 @@ lemma lsubf_refl: bi_reflexive … lsubf.
 /2 width=1 by lsubf_push, lsubf_bind/
 qed.
 
-lemma lsubf_refl_eq: â\88\80f1,f2,L. f1 â\89¡ f2 â\86\92 â¦\83L,f1â¦\84 â«\83ð\9d\90\85* â¦\83L,f2â¦\84.
+lemma lsubf_refl_eq: â\88\80f1,f2,L. f1 â\89¡ f2 â\86\92 â\9dªL,f1â\9d« â«\83ð\9d\90\85+ â\9dªL,f2â\9d«.
 /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/
-| @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *) 
+| @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *)
 ]
 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