partial commit in static_2

[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 e2ac985dc19f77c1e57615eb95eb1e71acbfe4c0..be342a26bd8ba9c31df6b3341a301bd67d4d5022 100644 (file)
--- 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: â\88\80f,f0,f1,f2,L1,L2,W,V. L1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ f → f0 ⋓ f ≘ f1 →
+              lsubf (L1.ⓘ[I]) (↑f1) (L2.ⓘ[I]) (↑f2)
+| lsubf_beta: â\88\80f,f0,f1,f2,L1,L2,W,V. L1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ f → f0 ⋓ f ≘ f1 →
               lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
-| lsubf_unit: â\88\80f,f0,f1,f2,I1,I2,L1,L2,V. L1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ f → f0 ⋓ f ≘ f1 →
-              lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
+| lsubf_unit: â\88\80f,f0,f1,f2,I1,I2,L1,L2,V. L1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ 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:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 → L1 = ⋆ →
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« → 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: â\88\80f1,f2,L2. â¦\83â\8b\86,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 → ∧∧ f1 ≡ f2 & 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« → ∧∧ f1 ≡ f2 & L2 = ⋆.
 /2 width=3 by lsubf_inv_atom1_aux/ qed-.
 
 fact lsubf_inv_push1_aux:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-     ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
-     â\88\83â\88\83g2,I2,K2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = ⫯g2 & L2 = K2.â\93\98{I2}.
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+     ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ[I1] →
+     â\88\83â\88\83g2,I2,K2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f2 = ⫯g2 & L2 = K2.â\93\98[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:
-      â\88\80g1,f2,I1,K1,L2. â¦\83K1.â\93\98{I1},⫯g1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-      â\88\83â\88\83g2,I2,K2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = ⫯g2 & L2 = K2.â\93\98{I2}.
+      â\88\80g1,f2,I1,K1,L2. â\9dªK1.â\93\98[I1],⫯g1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+      â\88\83â\88\83g2,I2,K2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f2 = ⫯g2 & L2 = K2.â\93\98[I2].
 /2 width=6 by lsubf_inv_push1_aux/ qed-.
 
 fact lsubf_inv_pair1_aux:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-     ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
-     â\88¨â\88¨ â\88\83â\88\83g2,K2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = â\86\91g2 & L2 = K2.â\93\91{I}X
-      | â\88\83â\88\83g,g0,g2,K2,W,V. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & 
-          K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+     ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ[I]X →
+     â\88¨â\88¨ â\88\83â\88\83g2,K2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f2 = â\86\91g2 & L2 = K2.â\93\91[I]X
+      | â\88\83â\88\83g,g0,g2,K2,W,V. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+          K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
           I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
-      | â\88\83â\88\83g,g0,g2,J,K2. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & 
-          K1 â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f2 = â\86\91g2 & L2 = K2.â\93¤{J}.
+      | â\88\83â\88\83g,g0,g2,J,K2. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+          K1 â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f2 = â\86\91g2 & L2 = K2.â\93¤[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:
-      â\88\80g1,f2,I,K1,L2,X. â¦\83K1.â\93\91{I}X,â\86\91g1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-      â\88¨â\88¨ â\88\83â\88\83g2,K2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = â\86\91g2 & L2 = K2.â\93\91{I}X
-       | â\88\83â\88\83g,g0,g2,K2,W,V. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & 
-           K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+      â\88\80g1,f2,I,K1,L2,X. â\9dªK1.â\93\91[I]X,â\86\91g1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+      â\88¨â\88¨ â\88\83â\88\83g2,K2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f2 = â\86\91g2 & L2 = K2.â\93\91[I]X
+       | â\88\83â\88\83g,g0,g2,K2,W,V. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+           K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
            I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
-       | â\88\83â\88\83g,g0,g2,J,K2. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & 
-           K1 â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f2 = â\86\91g2 & L2 = K2.â\93¤{J}.
+       | â\88\83â\88\83g,g0,g2,J,K2. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+           K1 â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f2 = â\86\91g2 & L2 = K2.â\93¤[J].
 /2 width=5 by lsubf_inv_pair1_aux/ qed-.
 
 fact lsubf_inv_unit1_aux:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-     ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
-     â\88\83â\88\83g2,K2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = â\86\91g2 & L2 = K2.â\93¤{I}.
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+     ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ[I] →
+     â\88\83â\88\83g2,K2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f2 = â\86\91g2 & L2 = K2.â\93¤[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:
-      â\88\80g1,f2,I,K1,L2. â¦\83K1.â\93¤{I},â\86\91g1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-      â\88\83â\88\83g2,K2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = â\86\91g2 & L2 = K2.â\93¤{I}.
+      â\88\80g1,f2,I,K1,L2. â\9dªK1.â\93¤[I],â\86\91g1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+      â\88\83â\88\83g2,K2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f2 = â\86\91g2 & L2 = K2.â\93¤[I].
 /2 width=5 by lsubf_inv_unit1_aux/ qed-.
 
 fact lsubf_inv_atom2_aux:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 → L2 = ⋆ →
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« → 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: â\88\80f1,f2,L1. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83â\8b\86,f2â¦\84 → ∧∧f1 ≡ f2 & 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« → ∧∧f1 ≡ f2 & L1 = ⋆.
 /2 width=3 by lsubf_inv_atom2_aux/ qed-.
 
 fact lsubf_inv_push2_aux:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-     ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
-     â\88\83â\88\83g1,I1,K1. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f1 = ⫯g1 & L1 = K1.â\93\98{I1}.
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+     ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ[I2] →
+     â\88\83â\88\83g1,I1,K1. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f1 = ⫯g1 & L1 = K1.â\93\98[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:
-      â\88\80f1,g2,I2,L1,K2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93\98{I2},⫯g2â¦\84 →
-      â\88\83â\88\83g1,I1,K1. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f1 = ⫯g1 & L1 = K1.â\93\98{I1}.
+      â\88\80f1,g2,I2,L1,K2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93\98[I2],⫯g2â\9d« →
+      â\88\83â\88\83g1,I1,K1. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f1 = ⫯g1 & L1 = K1.â\93\98[I1].
 /2 width=6 by lsubf_inv_push2_aux/ qed-.
 
 fact lsubf_inv_pair2_aux:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-     ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
-     â\88¨â\88¨ â\88\83â\88\83g1,K1. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f1 = â\86\91g1 & L1 = K1.â\93\91{I}W
-      | â\88\83â\88\83g,g0,g1,K1,V. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 &
-          K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+     ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ[I]W →
+     â\88¨â\88¨ â\88\83â\88\83g1,K1. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f1 = â\86\91g1 & L1 = K1.â\93\91[I]W
+      | â\88\83â\88\83g,g0,g1,K1,V. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+          K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ 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:
-      â\88\80f1,g2,I,L1,K2,W. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93\91{I}W,â\86\91g2â¦\84 →
-      â\88¨â\88¨ â\88\83â\88\83g1,K1. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f1 = â\86\91g1 & L1 = K1.â\93\91{I}W
-       | â\88\83â\88\83g,g0,g1,K1,V. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 &
-           K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+      â\88\80f1,g2,I,L1,K2,W. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93\91[I]W,â\86\91g2â\9d« →
+      â\88¨â\88¨ â\88\83â\88\83g1,K1. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f1 = â\86\91g1 & L1 = K1.â\93\91[I]W
+       | â\88\83â\88\83g,g0,g1,K1,V. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+           K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ 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:
-     â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
-     ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
-     â\88¨â\88¨ â\88\83â\88\83g1,K1. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f1 = â\86\91g1 & L1 = K1.â\93¤{I}
-      | â\88\83â\88\83g,g0,g1,J,K1,V. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & 
-          K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f1 = â\86\91g1 & L1 = K1.â\93\91{J}V.
+     â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
+     ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ[I] →
+     â\88¨â\88¨ â\88\83â\88\83g1,K1. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f1 = â\86\91g1 & L1 = K1.â\93¤[I]
+      | â\88\83â\88\83g,g0,g1,J,K1,V. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+          K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f1 = â\86\91g1 & L1 = K1.â\93\91[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:
-      â\88\80f1,g2,I,L1,K2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93¤{I},â\86\91g2â¦\84 →
-      â\88¨â\88¨ â\88\83â\88\83g1,K1. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f1 = â\86\91g1 & L1 = K1.â\93¤{I}
-       | â\88\83â\88\83g,g0,g1,J,K1,V. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & 
-           K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f1 = â\86\91g1 & L1 = K1.â\93\91{J}V.
+      â\88\80f1,g2,I,L1,K2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93¤[I],â\86\91g2â\9d« →
+      â\88¨â\88¨ â\88\83â\88\83g1,K1. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & f1 = â\86\91g1 & L1 = K1.â\93¤[I]
+       | â\88\83â\88\83g,g0,g1,J,K1,V. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« &
+           K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« â\89\98 g & g0 â\8b\93 g â\89\98 g1 & f1 = â\86\91g1 & L1 = K1.â\93\91[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:
-      â\88\80g1,f2,I1,I2,K1,K2. â¦\83K1.â\93\98{I1},⫯g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93\98{I2},f2â¦\84 →
-      â\88\83â\88\83g2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = ⫯g2.
+      â\88\80g1,f2,I1,I2,K1,K2. â\9dªK1.â\93\98[I1],⫯g1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93\98[I2],f2â\9d« →
+      â\88\83â\88\83g2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & 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:
-      â\88\80g1,f2,I,K1,K2. â¦\83K1.â\93\98{I},â\86\91g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93\98{I},f2â¦\84 →
-      â\88\83â\88\83g2. â¦\83K1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & f2 = ↑g2.
+      â\88\80g1,f2,I,K1,K2. â\9dªK1.â\93\98[I],â\86\91g1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93\98[I],f2â\9d« →
+      â\88\83â\88\83g2. â\9dªK1,g1â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & 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:
-      â\88\80g1,f2,K1,K2,V,W. â¦\83K1.â\93\93â\93\9dW.V,â\86\91g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93\9bW,f2â¦\84 →
-      â\88\83â\88\83g,g0,g2. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+      â\88\80g1,f2,K1,K2,V,W. â\9dªK1.â\93\93â\93\9dW.V,â\86\91g1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93\9bW,f2â\9d« →
+      â\88\83â\88\83g,g0,g2. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ 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:
-      â\88\80g1,f2,I,J,K1,K2,V. â¦\83K1.â\93\91{I}V,â\86\91g1â¦\84 â«\83ð\9d\90\85+ â¦\83K2.â\93¤{J},f2â¦\84 →
-      â\88\83â\88\83g,g0,g2. â¦\83K1,g0â¦\84 â«\83ð\9d\90\85+ â¦\83K2,g2â¦\84 & K1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+      â\88\80g1,f2,I,J,K1,K2,V. â\9dªK1.â\93\91[I]V,â\86\91g1â\9d« â«\83ð\9d\90\85+ â\9dªK2.â\93¤[J],f2â\9d« →
+      â\88\83â\88\83g,g0,g2. â\9dªK1,g0â\9d« â«\83ð\9d\90\85+ â\9dªK2,g2â\9d« & K1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ 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
@@ -253,123 +259,157 @@ lemma lsubf_inv_unit_sn:
 ]
 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
+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:
-      â\88\80f1,f2,I,L1,L2. â¦\83L1.â\93\98{I},f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2.â\93\98{I},f2â¦\84 â\86\92 â¦\83L1,⫱f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,⫱f2â¦\84.
+      â\88\80f1,f2,I,L1,L2. â\9dªL1.â\93\98[I],f1â\9d« â«\83ð\9d\90\85+ â\9dªL2.â\93\98[I],f2â\9d« â\86\92 â\9dªL1,â«°f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,â«°f2â\9d«.
 #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: 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/
-| #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: 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/
-| #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: 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/
+/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 *********************************************************)
 
-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. 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, 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/
+| #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, 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, 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-.
 
-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. 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, 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/
+| #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: ∀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: 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:
-      â\88\80g1,f2,I,L1,L2. â¦\83L1,g1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,⫱f2â¦\84 →
-      â\88\83â\88\83f1. â¦\83L1.â\93\98{I},f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2.â\93\98{I},f2â¦\84 & g1 = ⫱f1.
+      â\88\80g1,f2,I,L1,L2. â\9dªL1,g1â\9d« â«\83ð\9d\90\85+ â\9dªL2,â«°f2â\9d« →
+      â\88\83â\88\83f1. â\9dªL1.â\93\98[I],f1â\9d« â«\83ð\9d\90\85+ â\9dªL2.â\93\98[I],f2â\9d« & g1 = â«°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:
-      â\88\80f,f0,g1,L1,V. L1 â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ f → f0 ⋓ f ≘ g1 →
-      â\88\80f2,L2,W. â¦\83L1,f0â¦\84 â«\83ð\9d\90\85+ â¦\83L2,⫱f2â¦\84 →
-      â\88\83â\88\83f1. â¦\83L1.â\93\93â\93\9dW.V,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2.â\93\9bW,f2â¦\84 & ⫱f1 ⊆ g1.
+      â\88\80f,f0,g1,L1,V. L1 â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ f → f0 ⋓ f ≘ g1 →
+      â\88\80f2,L2,W. â\9dªL1,f0â\9d« â«\83ð\9d\90\85+ â\9dªL2,â«°f2â\9d« →
+      â\88\83â\88\83f1. â\9dªL1.â\93\93â\93\9dW.V,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2.â\93\9bW,f2â\9d« & â«°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 *) 
+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:
-      â\88\80f1,f2,L1,L2. â¦\83L1,f1â¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2â¦\84 →
+      â\88\80f1,f2,L1,L2. â\9dªL1,f1â\9d« â«\83ð\9d\90\85+ â\9dªL2,f2â\9d« →
       ∀f2l,f2r. f2l⋓f2r ≘ f2 →
-      â\88\83â\88\83f1l,f1r. â¦\83L1,f1lâ¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2lâ¦\84 & â¦\83L1,f1râ¦\84 â«\83ð\9d\90\85+ â¦\83L2,f2râ¦\84 & f1l⋓f1r ≘ f1.
+      â\88\83â\88\83f1l,f1r. â\9dªL1,f1lâ\9d« â«\83ð\9d\90\85+ â\9dªL2,f2lâ\9d« & â\9dªL1,f1râ\9d« â«\83ð\9d\90\85+ â\9dªL2,f2râ\9d« & 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-.