partial commit in static_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 2a6946fcab7ac6441c2109b34fbdfa422e9b625c..47912ace088d171406a07842e3ed6c36bcb6cb69 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma
@@ -19,7 +19,7 @@ include "static_2/syntax/lenv.ma".
 
 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
 
-inductive sex (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝
+inductive sex (RN,RP:relation3 lenv bind bind): pr_map → relation lenv ≝
 | sex_atom: ∀f. sex RN RP f (⋆) (⋆)
 | sex_next: ∀f,I1,I2,L1,L2.
             sex RN RP f L1 L2 → RN L1 I1 I2 →
@@ -37,7 +37,7 @@ 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 ≝
+           relation3 pr_map 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.
@@ -45,7 +45,7 @@ definition R_pw_transitive_sex:
 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 ≝
+           relation3 pr_map lenv bind ≝
            λR1,R2,RN,RP,f,L1,I1.
            ∀I2. R1 L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 → R2 L2 I1 I2.
 
@@ -53,7 +53,7 @@ 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 ≝
+           relation3 pr_map 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 →
@@ -63,7 +63,7 @@ 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 ≝
+           relation3 pr_map 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 →
@@ -87,9 +87,9 @@ fact sex_inv_next1_aux (RN) (RP):
      ∃∃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
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(eq_inv_pr_next_bi … H2) -g destruct
   /2 width=5 by ex3_2_intro/
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H)
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (eq_inv_pr_push_next … H)
 ]
 qed-.
 
@@ -104,8 +104,8 @@ fact sex_inv_push1_aux (RN) (RP):
      ∃∃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)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (eq_inv_pr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(eq_inv_pr_push_bi … H2) -g destruct
   /2 width=5 by ex3_2_intro/
 ]
 qed-.
@@ -131,9 +131,9 @@ fact sex_inv_next2_aux (RN) (RP):
      ∃∃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
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(eq_inv_pr_next_bi … H2) -g destruct
   /2 width=5 by ex3_2_intro/
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H)
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (eq_inv_pr_push_next … H)
 ]
 qed-.
 
@@ -148,8 +148,8 @@ fact sex_inv_push2_aux (RN) (RP):
      ∃∃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)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (eq_inv_pr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(eq_inv_pr_push_bi … H2) -g destruct
   /2 width=5 by ex3_2_intro/
 ]
 qed-.
@@ -180,7 +180,7 @@ 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) *
+#RN #RP #f #I1 #I2 #L2 #L2 elim (pr_map_split_tl f) *
 /2 width=1 by sex_next, sex_push/
 qed-.
 
@@ -190,14 +190,14 @@ 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 (pr_map_split_tl f) * #g #H destruct
 [ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
 qed-.
 
 (* Basic properties *********************************************************)
 
 lemma sex_eq_repl_back (RN) (RP):
-      ∀L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
+      ∀L1,L2. pr_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/
@@ -206,15 +206,15 @@ lemma sex_eq_repl_back (RN) (RP):
 qed-.
 
 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 *)
+      ∀L1,L2. pr_eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
+#RN #RP #L1 #L2 @pr_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).
 #RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
-#L #I #IH #f elim (pn_split f) *
+#L #I #IH #f elim (pr_map_split_tl f) *
 #g #H destruct /2 width=1 by sex_next, sex_push/
 qed.
 
@@ -246,8 +246,8 @@ lemma sex_co_isid (RN1) (RP1) (RN2) (RP2):
       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 //
-| /4 width=3 by sex_push, isid_inv_push/
+[ elim (pr_isi_inv_next … H) -H //
+| /4 width=3 by sex_push, pr_isi_inv_push/
 ]
 qed-.
 
@@ -257,9 +257,9 @@ lemma sex_sdj (RN) (RP):
       ∀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: // ]
+[ elim (pr_sdj_inv_next_sn … H12) -H12 [2,3: // ]
   #g2 #H #H2 destruct /3 width=1 by sex_push/
-| elim (sdj_inv_px … H12) -H12 [2,4: // ] *
+| elim (pr_sdj_inv_push_sn … H12) -H12 [2,4: // ] *
   #g2 #H #H2 destruct /3 width=1 by sex_next, sex_push/
 ]
 qed-.
@@ -270,10 +270,10 @@ lemma sle_sex_trans (RN) (RP):
       ∀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) * ]
-[ /4 width=5 by sex_push, sle_inv_pn/
-| /4 width=5 by sex_next, sle_inv_nn/
-| elim (sle_inv_xp … H12) -H12 [2,3: // ]
+[ elim (pr_map_split_tl f1) * ]
+[ /4 width=5 by sex_push, pr_sle_inv_push_next/
+| /4 width=5 by sex_next, pr_sle_inv_next_bi/
+| elim (pr_sle_inv_push_dx … H12) -H12 [2,3: // ]
   #g1 #H #H1 destruct /3 width=5 by sex_push/
 ]
 qed-.
@@ -284,10 +284,10 @@ lemma sle_sex_conf (RN) (RP):
       ∀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) * ]
-[ /4 width=5 by sex_push, sle_inv_pp/
-| /4 width=5 by sex_next, sle_inv_pn/
-| elim (sle_inv_nx … H12) -H12 [2,3: // ]
+[2: elim (pr_map_split_tl f2) * ]
+[ /4 width=5 by sex_push, pr_sle_inv_push_bi/
+| /4 width=5 by sex_next, pr_sle_inv_push_next/
+| elim (pr_sle_inv_next_sn … H12) -H12 [2,3: // ]
   #g2 #H #H2 destruct /3 width=5 by sex_next/
 ]
 qed-.
@@ -299,9 +299,9 @@ lemma sex_sle_split_sn (R1) (R2) (RP):
 #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
-[ elim (sle_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+[ elim (pr_sle_inv_next_sn … H ??) -H [ |*: // ] #g #Hfg #H destruct
   elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, ex2_intro/
-| elim (sle_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+| elim (pr_sle_inv_push_sn … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
   elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/
 ]
 qed-.
@@ -313,9 +313,9 @@ lemma sex_sdj_split_sn (R1) (R2) (RP):
 #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
-[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+[ elim (pr_sdj_inv_next_sn … H ??) -H [ |*: // ] #g #Hfg #H destruct
   elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/
-| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+| elim (pr_sdj_inv_push_sn … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
   elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/
 ]
 qed-.
@@ -332,7 +332,7 @@ lemma sex_dec (RN) (RP):
   lapply (sex_inv_atom2 … H) -H #H destruct
 | #L2 #I2 #f elim (IH L2 (⫰f)) -IH #HL12
   [2: /4 width=3 by sex_fwd_bind, or_intror/ ]
-  elim (pn_split f) * #g #H destruct
+  elim (pr_map_split_tl f) * #g #H destruct
   [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
   [1,3: /3 width=1 by sex_push, sex_next, or_introl/ ]
   @or_intror #H