X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fstatic%2Flsubf.ma;h=be342a26bd8ba9c31df6b3341a301bd67d4d5022;hp=49d435b3c7f78002a085a44232bf766834333f6a;hb=98e786e1a6bd7b621e37ba7cd4098d4a0a6f8278;hpb=c7b50fec51b9a25d5bc536f44e54179fd53efb44

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma b/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
index 49d435b3c..be342a26b 100644
--- a/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/lsubf.ma
@@ -12,22 +12,28 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground/xoa/ex_3_3.ma".
+include "ground/xoa/ex_4_3.ma".
+include "ground/xoa/ex_5_5.ma".
+include "ground/xoa/ex_5_6.ma".
+include "ground/xoa/ex_6_5.ma".
+include "ground/xoa/ex_7_6.ma".
 include "static_2/notation/relations/lrsubeqf_4.ma".
-include "ground_2/relocation/nstream_sor.ma".
+include "ground/relocation/nstream_sor.ma".
 include "static_2/static/frees.ma".
 
 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
 
-inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
+inductive lsubf: relation4 lenv pr_map lenv pr_map ≝
 | 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
@@ -37,7 +43,7 @@ interpretation
 (* Basic inversion lemmas ***************************************************)
 
 fact lsubf_inv_atom1_aux:
-     ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L1 = ⋆ →
+     ∀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/
@@ -48,80 +54,80 @@ fact lsubf_inv_atom1_aux:
 ]
 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}.
+     ∀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)
+  <(eq_inv_pr_push_bi … H1) -g1 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (eq_inv_pr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (eq_inv_pr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (eq_inv_pr_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}.
+      ∀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 &
+     ∀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}.
+      | ∃∃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)
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (eq_inv_pr_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/
+  <(eq_inv_pr_next_bi … 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/
+  <(eq_inv_pr_next_bi … 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/
+  <(eq_inv_pr_next_bi … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
 ]
 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 &
+      ∀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}.
+       | ∃∃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. ❪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 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (eq_inv_pr_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/
+  <(eq_inv_pr_next_bi … 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}.
+      ∀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,L2. ❪L1,f1❫ ⫃𝐅+ ❪L2,f2❫ → L2 = ⋆ →
      ∧∧ f1 ≡ f2 & L1 = ⋆.
 #f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
 [ /2 width=1 by conj/
@@ -132,94 +138,94 @@ fact lsubf_inv_atom2_aux:
 ]
 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}.
+     ∀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)
+  <(eq_inv_pr_push_bi … H1) -g2 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (eq_inv_pr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (eq_inv_pr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (eq_inv_pr_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}.
+      ∀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 &
+     ∀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)
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (eq_inv_pr_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/
+  <(eq_inv_pr_next_bi … 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/
+  <(eq_inv_pr_next_bi … 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,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 &
+      ∀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. ❪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 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (eq_inv_pr_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/
+  <(eq_inv_pr_next_bi … 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/
+  <(eq_inv_pr_next_bi … 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.
+      ∀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.
+      ∀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,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/
@@ -232,8 +238,8 @@ lemma lsubf_inv_bind_sn:
 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. ❪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
@@ -243,8 +249,8 @@ lemma lsubf_inv_beta_sn:
 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. ❪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
@@ -253,55 +259,55 @@ lemma lsubf_inv_unit_sn:
 ]
 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
+elim (pr_map_split_tl 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/
+#g2 #H12 #H destruct /3 width=5 by pr_eq_next, pr_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. ❪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 (pr_map_split_tl 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/
-| #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 //
+[ /2 width=3 by pr_isi_eq_repl_fwd/
+| /4 width=3 by pr_isi_inv_push, pr_isi_push/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (pr_isi_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (pr_isi_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (pr_isi_inv_next … H) -H //
 ]
 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/
-| #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 //
+[ /2 width=3 by pr_isi_eq_repl_back/
+| /4 width=3 by pr_isi_inv_push, pr_isi_push/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (pr_isi_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (pr_isi_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (pr_isi_inv_next … H) -H //
 ]
 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/
+/3 width=5 by pr_sor_inv_sle_sn_trans, pr_sle_next, pr_sle_push, pr_sle_refl_eq, pr_eq_sym/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
+lemma lsubf_eq_repl_back1: ∀f2,L1,L2. pr_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/
+  /3 width=3 by lsubf_atom, pr_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/
@@ -310,21 +316,21 @@ lemma lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃
   /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/
+  /3 width=5 by lsubf_beta, pr_sor_eq_repl_back/
 | #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/
+  /3 width=5 by lsubf_unit, pr_sor_eq_repl_back/
 ]
 qed-.
 
-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/
+lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. pr_eq_repl_fwd … (λf1. ❪L1,f1❫ ⫃𝐅+ ❪L2,f2❫).
+#f2 #L1 #L2 @pr_eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
 qed-.
 
-lemma lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
+lemma lsubf_eq_repl_back2: ∀f1,L1,L2. pr_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/
+  /3 width=3 by lsubf_atom, pr_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/
@@ -340,70 +346,70 @@ lemma lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃
 ]
 qed-.
 
-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/
+lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. pr_eq_repl_fwd … (λf2. ❪L1,f1❫ ⫃𝐅+ ❪L2,f2❫).
+#f1 #L1 #L2 @pr_eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
 qed-.
 
 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
+#L elim L -L /2 width=1 by lsubf_atom, pr_eq_refl/
+#L #I #IH #f elim (pr_map_split_tl f) * #g #H destruct
 /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.
+      ∀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
+elim (pr_map_split_tl 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.
+      ∀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/
+elim (pr_map_split_tl f2) * #x2 #H2 #L2 #W #HL12 destruct
+[ /3 width=4 by lsubf_push, pr_sor_inv_sle_sn, ex2_intro/
 | @(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⦄ →
+      ∀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.
+      ∃∃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/
+[ /3 width=7 by pr_sor_eq_repl_fwd, ex3_2_intro/
 | #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
-  elim (sor_inv_xxp … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
-  elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, sor_pp, ex3_2_intro/
+  elim (pr_sor_inv_push … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+  elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, pr_sor_push_bi, ex3_2_intro/
 | #g1 #g2 #I #L1 #L2 #_ #IH #f2l #f2r #H
-  elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
-  elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, sor_np, sor_pn, sor_nn, ex3_2_intro/
+  elim (pr_sor_inv_next … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+  elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, pr_sor_next_push, pr_sor_push_next, pr_sor_next_bi, ex3_2_intro/
 | #g #g0 #g1 #g2 #L1 #L2 #W #V #Hg #Hg1 #_ #IH #f2l #f2r #H
-  elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+  elim (pr_sor_inv_next … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
   elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
-  [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
-    /3 width=11 by lsubf_push, lsubf_beta, sor_np, ex3_2_intro/
-  | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
-    /3 width=11 by lsubf_push, lsubf_beta, sor_pn, ex3_2_intro/
-  | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
-    /3 width=11 by lsubf_beta, sor_nn, ex3_2_intro/
+  [ lapply (pr_sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
+    /3 width=11 by lsubf_push, lsubf_beta, pr_sor_next_push, ex3_2_intro/
+  | lapply (pr_sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
+    /3 width=11 by lsubf_push, lsubf_beta, pr_sor_push_next, ex3_2_intro/
+  | lapply (pr_sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
+    /3 width=11 by lsubf_beta, pr_sor_next_bi, ex3_2_intro/
   ]
 | #g #g0 #g1 #g2 #I1 #I2 #L1 #L2 #V #Hg #Hg1 #_ #IH #f2l #f2r #H
-  elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+  elim (pr_sor_inv_next … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
   elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
-  [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
-    /3 width=11 by lsubf_push, lsubf_unit, sor_np, ex3_2_intro/
-  | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
-    /3 width=11 by lsubf_push, lsubf_unit, sor_pn, ex3_2_intro/
-  | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
-    /3 width=11 by lsubf_unit, sor_nn, ex3_2_intro/
+  [ lapply (pr_sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
+    /3 width=11 by lsubf_push, lsubf_unit, pr_sor_next_push, ex3_2_intro/
+  | lapply (pr_sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
+    /3 width=11 by lsubf_push, lsubf_unit, pr_sor_push_next, ex3_2_intro/
+  | lapply (pr_sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
+    /3 width=11 by lsubf_unit, pr_sor_next_bi, ex3_2_intro/
   ]
 ]
 qed-.