update in staic_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / relocation / sex.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
index 3fba7476ad0d087ac365539b8668024fc793e1ee..72909212ca0f9bcbdc72968302e0932542730623 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
@@ -29,48 +29,62 @@ inductive sex (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝
             sex RN RP (⫯f) (L1.ⓘ[I1]) (L2.ⓘ[I2])
 .
 
-interpretation "generic entrywise extension (local environment)"
-   'Relation RN RP f L1 L2 = (sex RN RP f L1 L2).
-
-definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
-                           relation3 lenv bind bind →
-                           relation3 lenv bind bind → relation3 lenv bind bind →
-                           relation3 rtmap lenv bind ≝
-                           λR1,R2,R3,RN,RP,f,L1,I1.
-                           ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
-                           ∀I2. R2 L2 I I2 → R3 L1 I1 I2.
-
-definition R_pw_confluent2_sex: relation3 lenv bind bind → relation3 lenv bind bind →
-                                relation3 lenv bind bind → relation3 lenv bind bind →
-                                relation3 lenv bind bind → relation3 lenv bind bind →
-                                relation3 rtmap lenv bind ≝
-                                λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
-                                ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
-                                ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
-                                ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
-
-definition R_pw_replace3_sex: relation3 lenv bind bind → relation3 lenv bind bind →
-                              relation3 lenv bind bind → relation3 lenv bind bind →
-                              relation3 lenv bind bind → relation3 lenv bind bind →
-                              relation3 rtmap lenv bind ≝
-                              λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
-                              ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
-                              ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
-                              ∀I. R2 L1 I1 I → R1 L2 I2 I.
+interpretation
+  "generic entrywise extension (local environment)"
+  'Relation RN RP f L1 L2 = (sex RN RP f L1 L2).
+
+definition R_pw_transitive_sex:
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 lenv bind bind →
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 rtmap lenv bind ≝
+           λR1,R2,R3,RN,RP,f,L1,I1.
+           ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
+           ∀I2. R2 L2 I I2 → R3 L1 I1 I2.
+
+definition R_pw_confluent1_sex:
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 rtmap lenv bind ≝
+           λR1,R2,RN,RP,f,L1,I1.
+           ∀I2. R1 L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 → R2 L2 I1 I2.
+
+definition R_pw_confluent2_sex:
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 rtmap lenv bind ≝
+           λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
+           ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
+           ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
+           ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
+
+definition R_pw_replace3_sex:
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 lenv bind bind → relation3 lenv bind bind →
+           relation3 rtmap lenv bind ≝
+           λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
+           ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
+           ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
+           ∀I. R2 L1 I1 I → R1 L2 I2 I.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → X = ⋆ → Y = ⋆.
+fact sex_inv_atom1_aux (RN) (RP):
+     ∀f,X,Y. X ⪤[RN,RP,f] Y → X = ⋆ → Y = ⋆.
 #RN #RP #f #X #Y * -f -X -Y //
 #f #I1 #I2 #L1 #L2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆.
+lemma sex_inv_atom1 (RN) (RP):
+      ∀f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆.
 /2 width=6 by sex_inv_atom1_aux/ qed-.
 
-fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ[J1] → f = ↑g →
-                        ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2].
+fact sex_inv_next1_aux (RN) (RP):
+     ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ[J1] → f = ↑g →
+     ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2].
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J1 #K1 #H destruct
 | #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
@@ -80,12 +94,14 @@ fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ[J1] ⪤[RN,RP,↑g] Y →
-                     ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2].
+lemma sex_inv_next1 (RN) (RP):
+      ∀g,J1,K1,Y. K1.ⓘ[J1] ⪤[RN,RP,↑g] Y →
+      ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2].
 /2 width=7 by sex_inv_next1_aux/ qed-.
 
-fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ[J1] → f = ⫯g →
-                        ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2].
+fact sex_inv_push1_aux (RN) (RP):
+     ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ[J1] → f = ⫯g →
+     ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2].
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J1 #K1 #H destruct
 | #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
@@ -94,21 +110,25 @@ fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.
 ]
 qed-.
 
-lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ[J1] ⪤[RN,RP,⫯g] Y →
-                     ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2].
+lemma sex_inv_push1 (RN) (RP):
+      ∀g,J1,K1,Y. K1.ⓘ[J1] ⪤[RN,RP,⫯g] Y →
+      ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2].
 /2 width=7 by sex_inv_push1_aux/ qed-.
 
-fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → Y = ⋆ → X = ⋆.
+fact sex_inv_atom2_aux (RN) (RP):
+     ∀f,X,Y. X ⪤[RN,RP,f] Y → Y = ⋆ → X = ⋆.
 #RN #RP #f #X #Y * -f -X -Y //
 #f #I1 #I2 #L1 #L2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆.
+lemma sex_inv_atom2 (RN) (RP):
+      ∀f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆.
 /2 width=6 by sex_inv_atom2_aux/ qed-.
 
-fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ[J2] → f = ↑g →
-                        ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1].
+fact sex_inv_next2_aux (RN) (RP):
+     ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ[J2] → f = ↑g →
+     ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1].
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J2 #K2 #H destruct
 | #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
@@ -118,12 +138,14 @@ fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,↑g] K2.ⓘ[J2] →
-                     ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1].
+lemma sex_inv_next2 (RN) (RP):
+      ∀g,J2,X,K2. X ⪤[RN,RP,↑g] K2.ⓘ[J2] →
+      ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1].
 /2 width=7 by sex_inv_next2_aux/ qed-.
 
-fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ[J2] → f = ⫯g →
-                        ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1].
+fact sex_inv_push2_aux (RN) (RP):
+     ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ[J2] → f = ⫯g →
+     ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1].
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #J2 #K2 #g #H destruct
 | #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
@@ -132,37 +154,41 @@ fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.
 ]
 qed-.
 
-lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,⫯g] K2.ⓘ[J2] →
-                     ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1].
+lemma sex_inv_push2 (RN) (RP):
+      ∀g,J2,X,K2. X ⪤[RN,RP,⫯g] K2.ⓘ[J2] →
+      ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1].
 /2 width=7 by sex_inv_push2_aux/ qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair *)
-lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
-                    L1.ⓘ[I1] ⪤[RN,RP,↑f] L2.ⓘ[I2] →
-                    L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2.
+lemma sex_inv_next (RN) (RP):
+      ∀f,I1,I2,L1,L2.
+      L1.ⓘ[I1] ⪤[RN,RP,↑f] L2.ⓘ[I2] →
+      L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2.
 #RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H
 #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
-                    L1.ⓘ[I1] ⪤[RN,RP,⫯f] L2.ⓘ[I2] →
-                    L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2.
+lemma sex_inv_push (RN) (RP):
+      ∀f,I1,I2,L1,L2.
+      L1.ⓘ[I1] ⪤[RN,RP,⫯f] L2.ⓘ[I2] →
+      L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2.
 #RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H
 #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫱f] L2 →
-                  RN L1 I1 I2 → RP L1 I1 I2 →
-                  L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2].
+lemma sex_inv_tl (RN) (RP):
+      ∀f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫱f] L2 →
+      RN L1 I1 I2 → RP L1 I1 I2 →
+      L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2].
 #RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
 /2 width=1 by sex_next, sex_push/
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2.
-                    L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] →
-                    L1 ⪤[RN,RP,⫱f] L2.
+lemma sex_fwd_bind (RN) (RP):
+      ∀f,I1,I2,L1,L2.
+      L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] → L1 ⪤[RN,RP,⫱f] L2.
 #RN #RP #f #I1 #I2 #L1 #L2 #Hf
 elim (pn_split f) * #g #H destruct
 [ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
@@ -170,7 +196,8 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
+lemma sex_eq_repl_back (RN) (RP):
+      ∀L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
 #f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
 [ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/
@@ -178,40 +205,45 @@ lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L
 ]
 qed-.
 
-lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
+lemma sex_eq_repl_fwd (RN) (RP):
+      ∀L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
 #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *)
 qed-.
 
-lemma sex_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
-                ∀f.reflexive … (sex RN RP f).
+lemma sex_refl (RN) (RP):
+      c_reflexive … RN → c_reflexive … RP →
+      ∀f.reflexive … (sex RN RP f).
 #RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
 #L #I #IH #f elim (pn_split f) *
 #g #H destruct /2 width=1 by sex_next, sex_push/
 qed.
 
-lemma sex_sym: ∀RN,RP.
-               (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) →
-               (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) →
-               ∀f. symmetric … (sex RN RP f).
+lemma sex_sym (RN) (RP):
+      (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) →
+      (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) →
+      ∀f. symmetric … (sex RN RP f).
 #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f
 /3 width=2 by sex_next, sex_push/
 qed-.
 
-lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
-                     L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] →
-                     ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
-                     L1.ⓘ[J1] ⪤[RN,RP,f] L2.ⓘ[J2].
+lemma sex_pair_repl (RN) (RP):
+      ∀f,I1,I2,L1,L2.
+      L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] →
+      ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
+      L1.ⓘ[J1] ⪤[RN,RP,f] L2.ⓘ[J2].
 /3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-.
 
-lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
-              ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → L1 ⪤[RN2,RP2,f] L2.
+lemma sex_co (RN1) (RP1) (RN2) (RP2):
+      RN1 ⊆ RN2 → RP1 ⊆ RP2 →
+      ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → L1 ⪤[RN2,RP2,f] L2.
 #RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
 /3 width=1 by sex_atom, sex_next, sex_push/
 qed-.
 
-lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
-                   ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → 𝐈❪f❫ →
-                   L1 ⪤[RN2,RP2,f] L2.
+lemma sex_co_isid (RN1) (RP1) (RN2) (RP2):
+      RP1 ⊆ RP2 →
+      ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → 𝐈❪f❫ →
+      L1 ⪤[RN2,RP2,f] L2.
 #RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
 #f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
 [ elim (isid_inv_next … H) -H //
@@ -219,9 +251,10 @@ lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
 ]
 qed-.
 
-lemma sex_sdj: ∀RN,RP. RP ⊆ RN →
-               ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
-               ∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2.
+lemma sex_sdj (RN) (RP):
+      RP ⊆ RN →
+      ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+      ∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2.
 #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
 [ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
@@ -231,9 +264,10 @@ lemma sex_sdj: ∀RN,RP. RP ⊆ RN →
 ]
 qed-.
 
-lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP →
-                     ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
-                     ∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2.
+lemma sle_sex_trans (RN) (RP):
+      RN ⊆ RP →
+      ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+      ∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2.
 #RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
 #f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
 [ elim (pn_split f1) * ]
@@ -244,9 +278,10 @@ lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP →
 ]
 qed-.
 
-lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN →
-                    ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
-                    ∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2.
+lemma sle_sex_conf (RN) (RP):
+      RP ⊆ RN →
+      ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+      ∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2.
 #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
 [2: elim (pn_split f2) * ]
@@ -257,9 +292,10 @@ lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN →
 ]
 qed-.
 
-lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
-                     ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ⊆ g →
-                     ∃∃L. L1 ⪤[R1,RP,g] L & L ⪤[R2,cfull,f] L2.
+lemma sex_sle_split_sn (R1) (R2) (RP):
+      c_reflexive … R1 → c_reflexive … R2 →
+      ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ⊆ g →
+      ∃∃L. L1 ⪤[R1,RP,g] L & L ⪤[R2,cfull,f] L2.
 #R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
 [ /2 width=3 by sex_atom, ex2_intro/ ]
 #f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
@@ -270,9 +306,10 @@ lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
 ]
 qed-.
 
-lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
-                     ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ∥ g →
-                     ∃∃L. L1 ⪤[RP,R1,g] L & L ⪤[R2,cfull,f] L2.
+lemma sex_sdj_split_sn (R1) (R2) (RP):
+      c_reflexive … R1 → c_reflexive … R2 →
+      ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ∥ g →
+      ∃∃L. L1 ⪤[RP,R1,g] L & L ⪤[R2,cfull,f] L2.
 #R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
 [ /2 width=3 by sex_atom, ex2_intro/ ]
 #f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
@@ -283,10 +320,10 @@ lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
 ]
 qed-.
 
-lemma sex_dec: ∀RN,RP.
-               (∀L,I1,I2. Decidable (RN L I1 I2)) →
-               (∀L,I1,I2. Decidable (RP L I1 I2)) →
-               ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2).
+lemma sex_dec (RN) (RP):
+      (∀L,I1,I2. Decidable (RN L I1 I2)) →
+      (∀L,I1,I2. Decidable (RP L I1 I2)) →
+      ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2).
 #RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
 [ /2 width=1 by sex_atom, or_introl/
 | #L2 #I2 #f @or_intror #H