notational change for lexs

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lexs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
index e4e501f09471369bc6a4312c329046c978ef62c5..0b8465271c323f3a4e14dd4144ce7f451fd9d53a 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma
@@ -38,28 +38,28 @@ definition R_pw_confluent2_lexs: relation3 lenv term term → relation3 lenv ter
                                  relation3 rtmap lenv term ≝
                                  λR1,R2,RN1,RP1,RN2,RP2,f,L0,T0.
                                  ∀T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
-                                 â\88\80L1. L0 ⦻*[RN1, RP1, f] L1 â\86\92 â\88\80L2. L0 ⦻*[RN2, RP2, f] L2 →
+                                 â\88\80L1. L0 ⪤*[RN1, RP1, f] L1 â\86\92 â\88\80L2. L0 ⪤*[RN2, RP2, f] L2 →
                                  ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
 
 definition lexs_transitive: relation5 (relation3 lenv term term)
                                       (relation3 lenv term term) … ≝
                             λR1,R2,R3,RN,RP.
-                            â\88\80f,L1,T1,T. R1 L1 T1 T â\86\92 â\88\80L2. L1 ⦻*[RN, RP, f] L2 →
+                            â\88\80f,L1,T1,T. R1 L1 T1 T â\86\92 â\88\80L2. L1 ⪤*[RN, RP, f] L2 →
                             ∀T2. R2 L2 T T2 → R3 L1 T1 T2.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lexs_inv_atom1_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+fact lexs_inv_atom1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
 #RN #RP #f #X #Y * -f -X -Y //
 #f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: â\88\80RN,RP,f,Y. â\8b\86 ⦻*[RN, RP, f] Y → Y = ⋆.
+lemma lexs_inv_atom1: â\88\80RN,RP,f,Y. â\8b\86 ⪤*[RN, RP, f] Y → Y = ⋆.
 /2 width=6 by lexs_inv_atom1_aux/ qed-.
 
-fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g →
-                         â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g →
+                         â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J #K1 #W1 #H destruct
 | #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_next … H2) -g destruct
@@ -69,13 +69,13 @@ fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⦻*[RN, RP, ⫯g] Y →
-                      â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_next1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⪤*[RN, RP, ⫯g] Y →
+                      â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
 /2 width=7 by lexs_inv_next1_aux/ qed-.
 
 
-fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g →
-                         â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g →
+                         â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J #K1 #W1 #H destruct
 | #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #H elim (discr_next_push … H)
@@ -84,21 +84,21 @@ fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X
 ]
 qed-.
 
-lemma lexs_inv_push1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⦻*[RN, RP, ↑g] Y →
-                      â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_push1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⪤*[RN, RP, ↑g] Y →
+                      â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
 /2 width=7 by lexs_inv_push1_aux/ qed-.
 
-fact lexs_inv_atom2_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+fact lexs_inv_atom2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
 #RN #RP #f #X #Y * -f -X -Y //
 #f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: â\88\80RN,RP,f,X. X ⦻*[RN, RP, f] ⋆ → X = ⋆.
+lemma lexs_inv_atom2: â\88\80RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆.
 /2 width=6 by lexs_inv_atom2_aux/ qed-.
 
-fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g →
-                         â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g →
+                         â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #g #J #K2 #W2 #H destruct
 | #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_next … H2) -g destruct
@@ -108,12 +108,12 @@ fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y
 qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: â\88\80RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
-                      â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_next2: â\88\80RN,RP,g,J,X,K2,W2. X ⪤*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
+                      â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
 /2 width=7 by lexs_inv_next2_aux/ qed-.
 
-fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g →
-                         â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g →
+                         â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
 #RN #RP #f #X #Y * -f -X -Y
 [ #f #J #K2 #W2 #g #H destruct
 | #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #H elim (discr_next_push … H)
@@ -122,28 +122,28 @@ fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y
 ]
 qed-.
 
-lemma lexs_inv_push2: â\88\80RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 →
-                      â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_push2: â\88\80RN,RP,g,J,X,K2,W2. X ⪤*[RN, RP, ↑g] K2.ⓑ{J}W2 →
+                      â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
 /2 width=7 by lexs_inv_push2_aux/ qed-.
 
 (* Basic_2A1: includes lpx_sn_inv_pair *)
 lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2,V1,V2.
-                     L1.â\93\91{I1}V1 ⦻*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) →
-                     â\88§â\88§ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
+                     L1.â\93\91{I1}V1 ⪤*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) →
+                     â\88§â\88§ L1 ⪤*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
 #RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_next1 … H) -H
 #L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
 qed-.
 
 lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2,V1,V2.
-                     L1.â\93\91{I1}V1 ⦻*[RN, RP, ↑f] (L2.ⓑ{I2}V2) →
-                     â\88§â\88§ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
+                     L1.â\93\91{I1}V1 ⪤*[RN, RP, ↑f] (L2.ⓑ{I2}V2) →
+                     â\88§â\88§ L1 ⪤*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
 #RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_push1 … H) -H
 #L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
 qed-.
 
-lemma lexs_inv_tl: â\88\80RN,RP,f,I,L1,L2,V1,V2. L1 ⦻*[RN, RP, ⫱f] L2 →
+lemma lexs_inv_tl: â\88\80RN,RP,f,I,L1,L2,V1,V2. L1 ⪤*[RN, RP, ⫱f] L2 →
                    RN L1 V1 V2 → RP L1 V1 V2 → 
-                   L1.â\93\91{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2.
+                   L1.â\93\91{I}V1 ⪤*[RN, RP, f] L2.ⓑ{I}V2.
 #RN #RP #f #I #L2 #L2 #V1 #V2 elim (pn_split f) *
 /2 width=1 by lexs_next, lexs_push/
 qed-.
@@ -151,8 +151,8 @@ qed-.
 (* Basic forward lemmas *****************************************************)
 
 lemma lexs_fwd_pair: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. 
-                     L1.â\93\91{I1}V1 ⦻*[RN, RP, f] L2.ⓑ{I2}V2 →
-                     L1 ⦻*[RN, RP, ⫱f] L2 ∧ I1 = I2.
+                     L1.â\93\91{I1}V1 ⪤*[RN, RP, f] L2.ⓑ{I2}V2 →
+                     L1 ⪤*[RN, RP, ⫱f] L2 ∧ I1 = I2.
 #RN #RP #f #I1 #I2 #L2 #L2 #V1 #V2 #Hf
 elim (pn_split f) * #g #H destruct
 [ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf
@@ -161,7 +161,7 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lexs_eq_repl_back: â\88\80RN,RP,L1,L2. eq_repl_back â\80¦ (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_back: â\88\80RN,RP,L1,L2. eq_repl_back â\80¦ (λf. L1 ⪤*[RN, RP, f] L2).
 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
 #f1 #I #L1 #L2 #V1 #v2 #_ #HV #IH #f2 #H
 [ elim (eq_inv_nx … H) -H /3 width=3 by lexs_next/
@@ -169,7 +169,7 @@ lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⦻*[RN, RP,
 ]
 qed-.
 
-lemma lexs_eq_repl_fwd: â\88\80RN,RP,L1,L2. eq_repl_fwd â\80¦ (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_fwd: â\88\80RN,RP,L1,L2. eq_repl_fwd â\80¦ (λf. L1 ⪤*[RN, RP, f] L2).
 #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
 qed-.
 
@@ -191,9 +191,9 @@ lemma lexs_sym: ∀RN,RP.
 qed-.
 
 lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2.
-                      L1.â\93\91{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 →
+                      L1.â\93\91{I}V1 ⪤*[RN, RP, f] L2.ⓑ{I}V2 →
                       ∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 →
-                      L1.â\93\91{I}W1 ⦻*[RN, RP, f] L2.ⓑ{I}W2.
+                      L1.â\93\91{I}W1 ⪤*[RN, RP, f] L2.ⓑ{I}W2.
 #RN #RP #f #I #L1 #L2 #V1 #V2 #HL12 #W1 #W2 #HN #HP
 elim (lexs_fwd_pair … HL12) -HL12 /2 width=1 by lexs_inv_tl/
 qed-.
@@ -201,15 +201,15 @@ qed-.
 lemma lexs_co: ∀RN1,RP1,RN2,RP2.
                (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) →
                (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
-               â\88\80f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 â\86\92 L1 ⦻*[RN2, RP2, f] L2.
+               â\88\80f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 â\86\92 L1 ⪤*[RN2, RP2, f] L2.
 #RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
 /3 width=1 by lexs_atom, lexs_next, lexs_push/
 qed-.
 
 lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2.
                     (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
-                    â\88\80f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
-                    L1 ⦻*[RN2, RP2, f] L2.
+                    â\88\80f,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 #I #K1 #K2 #V1 #V2 #_ #HV12 #IH #H
 [ elim (isid_inv_next … H) -H //
@@ -218,8 +218,8 @@ lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2.
 qed-.
 
 lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
-                      â\88\80f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
-                      â\88\80f1. f1 â\8a\86 f2 â\86\92 L1 ⦻*[RN, RP, f1] L2.
+                      â\88\80f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+                      â\88\80f1. f1 â\8a\86 f2 â\86\92 L1 ⪤*[RN, RP, f1] L2.
 #RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
 #f2 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH
 [ * * [2: #n1 ] ] #f1 #H
@@ -231,8 +231,8 @@ lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
 qed-.
 
 lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) →
-                     â\88\80f1,L1,L2. L1 ⦻*[RN, RP, f1] L2 →
-                     â\88\80f2. f1 â\8a\86 f2 â\86\92 L1 ⦻*[RN, RP, f2] L2.
+                     â\88\80f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
+                     â\88\80f2. f1 â\8a\86 f2 â\86\92 L1 ⪤*[RN, RP, f2] L2.
 #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
 #f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH
 [2: * * [2: #n2 ] ] #f2 #H
@@ -244,8 +244,8 @@ lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) →
 qed-.
 
 lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                      â\88\80f,L1,L2. L1 ⦻*[R1, RP, f] L2 → ∀g. f ⊆ g →
-                      â\88\83â\88\83L. L1 ⦻*[R1, RP, g] L & L ⦻*[R2, cfull, f] L2.
+                      â\88\80f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ⊆ g →
+                      â\88\83â\88\83L. 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 lexs_atom, ex2_intro/ ]
 #f #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #y #H
@@ -259,7 +259,7 @@ qed-.
 lemma lexs_dec: ∀RN,RP.
                 (∀L,T1,T2. Decidable (RN L T1 T2)) →
                 (∀L,T1,T2. Decidable (RP L T1 T2)) →
-                â\88\80L1,L2,f. Decidable (L1 ⦻*[RN, RP, f] L2).
+                â\88\80L1,L2,f. Decidable (L1 ⪤*[RN, RP, f] L2).
 #RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #V1 #IH * ]
 [ /2 width=1 by lexs_atom, or_introl/
 | #L2 #I2 #V2 #f @or_intror #H