some restyling ...

[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 df09463c8085877bc9bdd7fc0c7ae95dcc99a1d9..6247b8686c3b92c04ea24ab3b1a33ed24bbc3ca3 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
@@ -38,7 +38,7 @@ definition R_pw_confluent2_sex: relation3 lenv bind bind → relation3 lenv 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 →
+                                ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
                                 ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
 
 definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
@@ -46,22 +46,22 @@ definition sex_transitive: 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 →
+                           ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
                            ∀I2. R2 L2 I I2 → R3 L1 I1 I2.
 
 (* 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
@@ -71,12 +71,12 @@ fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K
 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)
@@ -85,21 +85,21 @@ fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K
 ]
 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
@@ -109,12 +109,12 @@ fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K
 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)
@@ -123,28 +123,28 @@ fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K
 ]
 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.
+                    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.
+                    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 →
+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}.
+                  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-.
@@ -152,8 +152,8 @@ 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.
+                    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 //
@@ -161,7 +161,7 @@ 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/
@@ -169,7 +169,7 @@ lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f]
 ]
 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-.
 
@@ -189,20 +189,20 @@ lemma sex_sym: ∀RN,RP.
 qed-.
 
 lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
-                     L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
+                     L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} →
                      ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
-                     L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{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.
+              ∀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.
+                   ∀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 //
@@ -211,8 +211,8 @@ 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.
+               ∀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: // ]
@@ -223,8 +223,8 @@ 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.
+                     ∀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) * ]
@@ -236,8 +236,8 @@ 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.
+                    ∀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) * ]
@@ -249,8 +249,8 @@ 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.
+                     ∀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
@@ -262,8 +262,8 @@ 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.
+                     ∀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
@@ -277,7 +277,7 @@ 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).
+               ∀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