X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flsubf.ma;h=72e63576ecfdc41ce1ff019f9e779065e32cb5b4;hb=98fbba1b68d457807c73ebf70eb2a48696381da4;hp=1485963082fca335730675fccab7f2458506d37f;hpb=65e6209e0758832835ba8d14304a1548d059a634;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubf.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubf.ma
index 148596308..72e63576e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsubf.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubf.ma
@@ -18,11 +18,15 @@ include "basic_2/static/frees.ma".
 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
 
 inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
-| lsubf_atom: ∀f1,f2. f2 ⊆ f1 → lsubf (⋆) f1 (⋆) f2
-| lsubf_pair: ∀f1,f2,I,L1,L2,V. f2 ⊆ f1 → lsubf L1 (⫱f1) L2 (⫱f2) →
-              lsubf (L1.ⓑ{I}V) f1 (L2.ⓑ{I}V) f2
-| lsubf_beta: ∀f,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≡ f → f ⊆ ⫱f1 → f2 ⊆ f1 →
-              lsubf L1 (⫱f1) L2 (⫱f2) → lsubf (L1.ⓓⓝW.V) f1 (L2.ⓛW) f2
+| 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_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 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)
 .
 
 interpretation
@@ -32,87 +36,278 @@ interpretation
 (* Basic inversion lemmas ***************************************************)
 
 fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
-                          L2 = ⋆ ∧ f2 ⊆ f1.
+                          f1 ≗ f2 ∧ L2 = ⋆.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ /2 width=1 by conj/
-| #f1 #f2 #I #L1 #L2 #V #_ #_ #H destruct
-| #f #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
+| #f1 #f2 #I #L1 #L2 #_ #H destruct
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ ∧ f2 ⊆ f1.
+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}.
+#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
+  <(injective_push … H1) -g1 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+]
+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}.
+/2 width=6 by lsubf_inv_push1_aux/ qed-.
+
 fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
-                          ∀I,K1,X. L1 = K1.ⓑ{I}X →
-                          (∃∃K2. f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L2 = K2.ⓑ{I}X) ∨
-                          ∃∃f,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
-                                      f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+                          ∀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 #_ #J #K1 #X #H destruct
-| #f1 #f2 #I #L1 #L2 #V #H21 #HL12 #J #K1 #X #H destruct
-  /3 width=3 by ex3_intro, or_introl/
-| #f #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H21 #HL12 #J #K1 #X #H destruct
-  /3 width=11 by ex7_4_intro, or_intror/
+[ #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)
+| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+  <(injective_next … H1) -g1 /3 width=5 by or3_intro0, ex3_2_intro/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+  <(injective_next … H1) -g1 /3 width=12 by or3_intro1, ex7_6_intro/
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+  <(injective_next … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
 ]
 qed-.
 
-lemma lsubf_inv_pair1: ∀f1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
-                       (∃∃K2. f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L2 = K2.ⓑ{I}X) ∨
-                       ∃∃f,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
-                                   f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-/2 width=3 by lsubf_inv_pair1_aux/ 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}.
+/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}.
+#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)
+| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #H1 #H2 destruct
+  <(injective_next … H1) -g1 /2 width=5 by ex3_2_intro/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
+]
+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}.
+/2 width=5 by lsubf_inv_unit1_aux/ qed-.
 
 fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
-                          L1 = ⋆ ∧ f2 ⊆ f1.
+                          f1 ≗ f2 ∧ L1 = ⋆.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ /2 width=1 by conj/
-| #f1 #f2 #I #L1 #L2 #V #_ #_ #H destruct
-| #f #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
+| #f1 #f2 #I #L1 #L2 #_ #H destruct
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → L1 = ⋆ ∧ f2 ⊆ f1.
+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}.
+#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
+  <(injective_push … H1) -g2 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+]
+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}.
+/2 width=6 by lsubf_inv_push2_aux/ qed-.
+
 fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
-                          ∀I,K2,W. L2 = K2.ⓑ{I}W →
-                          (∃∃K1.f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L1 = K1.ⓑ{I}W) ∨
-                          ∃∃f,K1,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
-                                    f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abst & L1 = K1.ⓓⓝW.V.
+                          ∀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 #_ #J #K2 #X #H destruct
-| #f1 #f2 #I #L1 #L2 #V #H21 #HL12 #J #K2 #X #H destruct
-  /3 width=3 by ex3_intro, or_introl/
-| #f #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H21 #HL12 #J #K2 #X #H destruct
-  /3 width=7 by ex6_3_intro, or_intror/
+[ #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)
+| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #X #H1 #H2 destruct
+  <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g2 #J #K2 #X #H1 #H2 destruct
+  <(injective_next … H1) -g2 /3 width=10 by ex6_5_intro, or_intror/
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J #K2 #X #_ #H destruct
 ]
 qed-.
 
-lemma lsubf_inv_pair2: ∀f1,f2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, f2⦄ →
-                       (∃∃K1.f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L1 = K1.ⓑ{I}W) ∨
-                       ∃∃f,K1,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⊆ ⫱f1 &
-                                 f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & 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.
+#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)
+| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #H1 #H2 destruct
+  <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J #K2 #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g2 #J #K2 #H1 #H2 destruct
+  <(injective_next … H1) -g2 /3 width=11 by ex5_6_intro, or_intror/
+]
+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.
+/2 width=5 by lsubf_inv_unit2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+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.
+#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.
+#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/
+  | #z #z0 #z2 #Y2 #W #V #_ #_ #_ #_ #H0 #_ #H destruct
+  | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
+  ]
+| elim (lsubf_inv_unit1 … H) -H
+  #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
+]
+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.
+#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
+  /2 width=7 by ex4_3_intro/
+| #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
+]
+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.
+#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
+| #z #z0 #z2 #Z2 #Y2 #H02 #Hz #Hg1 #H0 #H1 destruct
+  /2 width=7 by ex4_3_intro/
+]
+qed-.
+
+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
+[ elim (lsubf_inv_push_sn … H12) | elim (lsubf_inv_bind_sn … H12) ] -H12
+#g2 #H12 #H destruct /3 width=5 by eq_next, eq_push/
+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⦄.
+#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⦄.
+#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/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+]
+qed-.
+
+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/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+]
+qed-.
+
 lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
-#f1 #f2 #L1 #L2 * //
+#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 *********************************************************)
 
-lemma lsubf_pair_nn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
-                     ∀I,V. ⦃L1.ⓑ{I}V, ⫯f1⦄ ⫃𝐅* ⦃L2.ⓑ{I}V, ⫯f2⦄.
-/4 width=5 by lsubf_fwd_sle, lsubf_pair, sle_next/ qed.
+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⦄).
+#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_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-.
 
-lemma lsubf_refl: ∀L,f1,f2. f2 ⊆ f1 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
-#L elim L -L /4 width=1 by lsubf_atom, lsubf_pair, sle_tl/
+lemma lsubf_refl: bi_reflexive … lsubf.
+#L elim L -L /2 width=1 by lsubf_atom, eq_refl/
+#L #I #IH #f elim (pn_split f) * #g #H destruct
+/2 width=1 by lsubf_push, lsubf_bind/
 qed.
 
-lemma lsubf_sle_div: ∀f,f2,L1,L2. ⦃L1, f⦄ ⫃ 𝐅* ⦃L2, f2⦄ →
-                     ∀f1. f1 ⊆ f2 → ⦃L1, f⦄ ⫃ 𝐅* ⦃L2, f1⦄.
-#f #f2 #L1 #L2 #H elim H -f -f2 -L1 -L2
-/4 width=3 by lsubf_beta, lsubf_pair, lsubf_atom, sle_tl, sle_trans/
+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_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-.
+